Series Puzzle

Have a look at these two sequences.

120
222
333
444
555
675
777
897
9117
10119
111210
121410
131610
141612
151614
161814
172014
182115
192216
202317
212319
222321
232323
242523

* * *

120
240
360
471
582
6102
7122
8133
9144
10164
11175
12195
13215
14226
15237
16257

* * *

Now, have a look at these wrong sequences, which are malformed according to the rules underlying the sequence.

120
222
333
444
555
675
777
897
9117
10119
111210
111311
121511
131711
141812
151814
162014
172016
182117
192218
202319
212321
222521
232523
242624

* * *

120
222
333
444
555
675
777
897
9117
10119
111210
121410
131610
141612
151614
161814
172014
182115
192216
202317
212319
222321
232424
242624
* * *

A LLM could not solve this problem on its own, requiring many hints before it finally stumbled across the correct answer. Can you do better?

Here are some hints for you, fair reader. If you can complete it without using all of them, you have done better than the bot. If you can complete it without using any, you’re either a puzzle solving mastermind or very in tune with how I think. Each hint is broken up by a line, like so: ^ ^ ^

^ ^ ^ ^ ^ ^ ^

Hint #1: It’s not a traditional mathematical number sequence – that is, the number as a whole doesn’t progress by a single rule which is consistently applied from one number to the next. It consists of multiple variables, two of which are affected by a single external (undescribed, but predictable when you understand the incrementation) condition.

^ ^ ^ ^ ^ ^ ^

Hint #2: The sequence terminates because of an external condition.

^ ^ ^ ^ ^ ^ ^

Hint #3: The chief external condition that controls the middle and end values, from a mathematical point of view, a variable with three possible states. It can vary from row to row.

^ ^ ^ ^ ^ ^ ^
CONTEXT CLUE COMING UP…
^ ^ ^ ^ ^ ^ ^
Hint #4: board game competitions.

Gomez Puzzle

Señor Gomez is an eccentric person and has very strong opinions about certain things:
He likes skirts but hates dresses.
He likes computers but hates mobile phones.
He likes motorcycles but hates cars.
He likes tables but hates desks.
He likes jackets but hates coats.
He likes mugs but hates glasses.
He likes cards but hates cash.
He likes tortillas but hates bread.

All of his friends are confused. Can you understand his preferences?

Announcing Focus and Launch Menus

I’m pleased to announce the release of two new Xfce Panel applets, Focus Menu and Launch Menu, which reproduce functionality that existed on vintage Apple computers, from System 7 up to Mac OS 9.

Focus Menu works similarly to the application switcher menu, displaying a list of programs, as well as commands that allow their windows to be hidden or shown. Launch Menu works similarly to the Apple menu, showing a list of files, folders and programs that the user manually placed into the special Launch Menu Items folders.

Read about each release, and also my attempts to anticipate all the common questions, on their respective GitHub pages:

Focus Menu

Launch Menu

(Yes, I know… the retirement didn’t last long.)

N-in-a-Row Games, Part 3

This is a classic gPress post, originally uploaded October 21st, 2022.

I looked for a great Tic-Tac-Toe variant, and managed to stumble across it, thanks to the famous article on Math with Bad Drawings.

Mr. Orlin’s description of the rules of the game have no obvious room for improvement, so instead of taking the time to write a worse description, this post will build on his foundation to describe terms, openings, materials, and other items that I’ve found useful as a player.

Table of Contents
Glossary of Terms
Playing Materials
Misc. Norms, Tips, et al.
General Strategy
Openings

Glossary of Terms

The game proper is referred to long-form as “Ultimate Tic-Tac-Toe”, or as merely “Ultimate”, but we’ve taken to calling it “ULT“, a short form which distinguishes from other games (like the frisbee tossing game).

A square that keeps the play on the same board (for instance, the center square on the center board) is called a ‘home square‘.

The ability to place a piece on any board is referred to as a ‘free move‘, and a square that allows your opponent to perform this move can be called a ‘free move square‘.

There are several means of notation that I have discovered or thought of:
1) Algebraic: Like algebraic notation on a chess board, with letters corresponding to columns and numbers to rows. The bottom-left square is a1, the center square is e5, and the top-right is i9.
2) Compass: Uses directions like “NW”, “SE”, or “C”. The board picked is first, then the square on the board. The bottom-left square is SW/SW, the center is C/C, and the top-right is NE/NE.
3) Numeric : Each square has a number. The first number is the board you are playing on, the second is the square you are moving on. The orientation may vary depending on whether you are using a telephone-style keypad or a computer/calculator-style keypad; the latter is my default. The bottom-left square is 11, the center is 55, and the top-right is 99.
4) Relative : Like the compass system, but uses arrow symbols instead of letters. This can help in-person play, because the arrows will be correct from each player’s perspective. Bottom-left would be ↙↙, center would be ••, and top-right would be ↗↗.

I prefer the numeric system in principle, but my games tend to be written in algebraic notation, because I was familiar with it from chess and didn’t think of a better way to do it until after I had a large body of work in the former.

For denoting a move that wins the game, or trivially forces a win, the convention is to use # following your move, the same as in chess checkmate.

This is not standard notation, but a move that captures a board may be denoted by putting ‘[X]’ (or perhaps ‘[O]’) at the end of it. Likewise, a move that causes a free move may be noted by “FM”. If another symbol is chosen, I would recommend an asterisk [*] or perhaps an at symbol [@].

Yu can refer to a move that captures a board as being ‘3r’, and a move that wins the game as being ‘3br’.

Playing Materials

In online play, I use a spreadsheet and exchange moves with my opponent via IRC or a similar protocol. I developed a custom spreadsheet some time back, which helps keep track of free move squares, allows convenient text input of moves, and can display captured boards. Click here to download it for yourself. The first page also includes match details, useful for a series of games.

There are several computer implementations that manage the board and can help arrange online games. The most popular one used to be Bennett Zhang’s Heroku app, but it seems to be offline, as of 9/1/23. Arkadiusz Nowaczyński’s uttt.ai includes useful analysis options, as well as probably the strongest AI that exists, which plays as well as the strongest chess computers play chess. Ofek Gila’s program is older, but also features a strong AI.

For in-person play, you can either use pen-and-paper, or manufacture a board and pieces. If you don’t mind using printer ink, you can expedite the process of board-making a little bit by printing out “Game Gal’s” boards.

Misc. Norms, Tips, et al.

A player familiar with the game is able to play at a ‘classical’ level in 25-45 minutes. Two games in 60-70 minutes is a decent pace with sufficient time for thought. If a game goes on longer than an hour, someone is taking up too much time. A game shouldn’t go on as long as an hour and a half.

As in regular Tic-Tac-Toe, the first mover (by convention, X) has an advantage. Like Chess, the game is not proven to be a draw with perfect play, but it likely is. In a fair match, each player should have an equal number of games as first and second mover.

Mr. Orlin mentions a variation of the game where ‘tie’ boards are considered won for both players (whereas the norm is for those boards to be considered inert). We played this several times as “Cutthroat” (or “CUT”). Most games seem to be unaffected, but there are positions where this makes all the difference. I would stick with ‘inert’ boards for ties, on the grounds that it is mainstream and more intuitive. If you do play CUT, the norm we figured is that in the case of simultaneous opposing 3brs, the game is drawn.

Another variation is known where the play is the same, but the objective of the game is to win 5 boards. This does affect the outcome of the game, for similar reasons as why a popular vote can result in different outcomes from the electoral college. While it is ULT-like, I would not call this game “Ultimate” or “ULT”. The game seems to have been named “Tic-Tac-Ku”, and I would stick with that, for the purpose of distinguishing the two.

General Strategy

Several articles could be written on this subject, and perhaps will be in the future, but here I will try to give a brief overview.

This game is a new frontier in many ways, and there’s less known about play and principles of play than I would like. We are all students of the game, and it is best to avoid being dogmatic about an approach. A type of move that works in one case may turn out to be a disaster in another. Any advice given can only be very general, subject to circumstances.

From our earlier study of tic-tac-toe, we know that the center square is mathematically the strongest square on a tic-tac-toe board. Therefore, we understand that the center board is the strongest board in ULT. The difference is that while the center square is easily defended against in a tic-tac-toe game, good positioning on the center board can provide an enduring advantage that is difficult to offset.

Whenever you move on a square, you are giving your opponent compensating value in the board that you’re directing them to. A major principle of play is to get gain in boards you care about while trying to make sure your opponent only gets advantages in boards that you don’t. This is complicated by the possibility of your moves being ‘wasted’ by overplaying on a board that you already have a winning advantage in, or by being forced to play in boards that will lead to a good gain for your opponent.

As boards are won and the game gets to the midgame or endgame, there will be many times when you don’t want to move on a certain board, because the only squares you can move on there will give your opponent the chance to make a winning play on another board. From my experience, these positions are seldom clear in advance, and can take up calculation time to uncover.

Never underestimate the power of a free move. Many games have been won by someone being forced to move on a free move square. Don’t give up free moves for free.

On the other hand, don’t underestimate the power of a good major board position, even if securing it gives up a free move. If your opponent has a 3br threat, it is often worth quenching it, even if that gives them a free move somewhere else. A 3br threat, once it exists, will hang over your head for the rest of the game, because it means that any free move square you land on ends the game. Likewise, you want to have a good 3br threat if you can get one. It’s a move that pays you interest.

If you can win or draw on merely three boards – the center, and two opposing corners – that is sufficient to not lose the game, even if your opponent wins every other board. (This is why the outcome of Tic-Tac-Ku is so different from regular ULT.)

Because we can’t see everything, it is often a good idea to ‘hedge your bets’ and not commit to trying to win a certain line of boards (at the expense of the others) too early. A board that seems to have a bad position one turn may turn out to be useful just a few minutes later.

Once boards start being won, it is harder to win others without paying an extra cost. The sheer mass of your opponent’s already-won boards, even if they are not particularly good boards, can make all the difference when you’re both being forced into tight spaces.

Openings

Because the board is symmetrical, there are fewer openings than there are squares:
Center-Center (e5, 55) – “The Main Line”
Center-Corner (d4/51, d6/57, etc.) – “The Corner Attack”
Center-Edge (e4/52, e6/58, etc.)
Corner-Home Corner (a1/11, a9/77, etc.)
Corner-Alternate Corner (a3/17)
Corner-Opposing Corner (c3/19, etc)
Corner-Center (a2/15, etc.)
Corner-Aligned Edge (a2/12)
Corner-Unaligned Edge (b3/18)
… and a plethora of edge board openings too tedious to mention in detail.
UTTT.ai analysis scores are shown after an opening’s name.

Center-Center (e5, 55) – The Center Game – “The Main Line” – +11.81
This is considered the most solid opening. It gives you a decent (though not insurmountable) advantage on the center board, and your opponent’s move will give you the option of playing the first move on a second board, probably a corner board. Strong AIs strongly prefer this move.

Center-Corner (d4/51, d6/57, etc.) – The Corner Game – “The Corner Attack” – +11.07
When it comes to alternate openings (besides e5), this was my initial unstudied preference. This move seems to give a greater edge on the center board than e5 does, because of the lack of a piece to oppose your own. I second-guessed this opening for a while, but computer analysis has revealed that it is, in fact, a top-2 opening, as I originally imagined.

Center-Edge (e4/52, e6/58, etc.) – +8.16
The center-edge opening places an uncontested piece on the center board and gives O the lead move in an edge board. The edge square is weak in the center board, because it has only two lines, one of which runs through the center square – which is also weaker than normal because it allows the opponent to immediately reply. I’d suspect I’m not meeting my potential here, but it is likely playable, since the edge board compensation your opponent gets is slim.

Corner-Home Corner (a1/11, a9/77, etc.) – +6.47
Places a contested piece on a corner board, giving X and O near-equality on this board and allowing them to pick the next board, but keeping the first move on that board. This move is not obviously bad and seems, at first glance, not worse than the following options.

Corner-Alternate Corner (a3/17)
Places an uncontested piece on a corner board, allowing O the lead move in another corner. This move completely ignores the center board, leaving that board’s status the ‘elephant in the room’. This is perhaps one of the purest openings in terms of allowing the player to measure the strength of the first-move advantage, as X and O’s positions at this point seem rather equal otherwise.

Corner-Opposing Corner (c3/19, etc) – +6.12
Places an uncontested piece on a corner board, allowing O the lead move in the opposing corner. This move is similar to the previous one, but has the characteristic of weakening your chances for 3br from your initial board, due to giving compensation to your opponent at the other end of the line. I am unsure of it on those grounds.

Corner-Unaligned Edge (b3/18)
Places an uncontested piece on a corner board, allowing O the first move on an edge board that does not overlap with the corner board’s rows. It seems plausible that X maintains a slight advantage here and that the move is not outright ‘bad’.

Corner-Aligned Edge (a2/12)
Places an uncontested piece on a corner board, allowing O the first move on an edge board that overlaps with the corner board’s rows. My intuitive guess is that this position is a little worse than the previous one, although it is hard to measure or project without an in-depth study.

Corner-Center (a2/15, etc.)
Places an uncontested piece on a corner board and gives O the lead move in the center board, which gives them similar prerogatives to the main openings. This move seems to put O in the driver’s seat, allowing X to defend with a position stronger than what O could have had. Philosophically I doubt that it is as viable an opening as the others, but it might be a good psychological ploy, or a good move for someone who is used to playing as the second player and would like to reuse as much of their knowledge as possible.

Edge Board Openings
There is a wide variety of possible openings here, given that there’s not only a home-square edge, but also corners or edges that do not overlap with the first board’s rows, versus ones that do, but they seem to have one thing in common: they’re all bad. Any move on an edge board is likely to either result in a seemingly equal position (not clearly accomplishing anything for X), or to even give O the advantage. This might be viable for a handicap game or a pie-rule tiebreaker, but it should be ruled out as a serious move between comparable players.

Conclusion

Check back periodically to see if there have been revisions to this, as there likely will be in the future. In the meantime, be on the lookout for a fourth article; there is still more to say about this game, and also about some close variants of it.

 

N-in-a-Row Games, Parts 1 & 2

These are two classic gPress posts, originally uploaded August 28, 2022 and September 2, 2022 respectively, combined into one.

Some years back, I challenged a friend to a long (100-game) series to determine who would be the World Champion Tic-Tac-Toe player. What started as a jest developed into a serious interest into the world of n-in-a-row games. In these posts, I over-analyze simple things, show the depth of the better games, and try to instill an interest in them.

Tic-Tac-Toe, three in a row…

Tic-Tac-Toe is proven to be a draw with perfect play, but it is a good starting point to understand the principles of n-in-a-row games.

As everyone in the western world knows, in a regular game of Tic-Tac-Toe, there are nine squares, in three rows and columns. Each player, X and O, take turns placing one of their pieces on an unoccupied square. If there is three-in-a-row of one piece, that player wins. Otherwise the game is a draw (or “cat” game).

Breaking it down more mathematically, we see that on a three-by-three (3×3) Tic-Tac-Toe board, there are eight possible rows of three: the two diagonals, three horizontal, and three vertical. Because the board layout is symmetrical, each corner and edge square initially have identical value. This also means that there are only three openings: center, corner, and edge. In general, a square that has more possible n-in-a-rows through it is a more powerful square, so the center square, which can be in four possible rows, would seem to be the strongest starting position.

a b c
3
2 X
1

In this situation, O has two options, a corner or an edge. After subtracting the rows that are made unwinnable due to the first player’s move, any corner has two rows that go through it; any edge has only one, so the corner is mathematically and intuitively the stronger square.

(If we play the edge, we find that the edge defense loses; X plays either a corner or an edge diagonal to the edge chosen, forcing O to defend against 3r, then X can immediately set up a 3r threat from both of his previous squares.

Center Game, Edge Blunder, Corner Variation
1. b2 a2
2. a1+ c3
3. c1#

a b c
3 O
2 O X
1 X X

On the other hand, after the corner move, although X is in the stronger position and has the initiative, he is usually unable to win. Most moves he makes will set up a 3r threat, but can also be immediately and directly countered by O.

Center Game, Corner Line
1. b2 a1
2. a3+ c1+
3. b1+ b3
4. c2+ a2~ (draw)

a b c
3 X O
2 O X X
1 O X O

X does have one possible option for a trap:

Center Game, Trap Sprung
1. b2 a1
2. c3 a2+
3. a3#

a b c
3 X X
2 O X
1 O

The proper defense is to play on a corner.

Center Game, Trap Avoided
1. b2 a1
2. c3 c1+
3. b1+ b3 … and draw

a b c
3 O X
2 X
1 O X O

This sums up the center game.

Because of the simplicity of the defense against it, the center opening is sometimes eschewed in favor of a corner attack.

a b c
3
2
1 X

Consider the position O is in. A corner is taken, meaning that there are only five out of the eight possible rows that O can still win. The center now has three such paths; each remaining corner has two. The edges orthogonal to the occupied corner have one, but the others still have two.

The corner and edge defenses are fatally flawed in a similar fashion to the edge defense previously discussed.

Corner Game, Opposite Corner Blunder
1. a1 c3
2. c1+ c2
3. a3#

a b c
3 X O
2
1 X O X

Corner Game, Adjacent Corner Blunder
1. a1 c1
2. c3 b2
3. a3#

a b c
3 X X
2 O
1 X O

Corner Game, Opposite Edge Blunder
1. a1 b3
2. a3+ b2
3. c1#

a b c
3 X O
2 O
1 X X

Corner Game, Adjacent Edge Blunder
1. a1 a2
2. b2+ a3
3. c1#

a b c
3 O
2 O X
1 X X

So, O moves in the center.

a b c
3
2 O
1 X

Because of O’s move, X’s square is now worse than it was previously, and is actually worth a little less than O’s. Having the first move advantage means he keeps the momentum, but this is only enough to guarantee a draw. Why, then, does anyone prefer the corner?

If O isn’t careful, she may wander into a trap.

The favorite X response is normally to play the corner opposite to his initial move, which sets up the corner trap: if O moves in one of the other corners, X takes the third corner, blocking 3r and creating two 3r threats of his own, winning the game.

Corner Game, Corner Trap Sprung
1. a1 b2
2. c3 a3+
3. c1#

a b c
3 O X
2 O
1 X X

A more subtle but similar variation is the edge trap. X plays in one of the edges adjacent to the opposite corner. If O plays in the wrong corner (the corner furthest from the other two squares), X can again block her 3r attempt and create two threats of his own.

Center Game, Edge Trap Sprung
1. a1 b2
2. b3 c1+
3. a3#

a b c
3 X X
2 O
1 X O

However, both of these traps require specific wrong moves from O to succeed. With other easily found moves, the game will end with a draw after a series of blocked 3r attempts. Some games may run out of possible rows before they run out of moves.

Corner Game, Corner Trap Avoided
1. a1 b2
2. c3 a2+
3. c2+ c1+
4. a3+ b3+
5. b1 and draw

a b c
3 X O X
2 O O X
1 X X O

Corner Game, Edge Trap Avoided
1. a1 b2
2. b3 a3+
3. c1+ b1 and an easy draw

a b c
3 O X
2 O
1 X O X

Some players have criticized the edge opening and claimed that it loses outright. This is false, but it does require more precise play from X than the other lines. However, because some players are unfamiliar with this system, this may be a way to try for a win.

There are unsound defenses from O that lose by force. These are the adjacent edge and opposing corner.

Edge Game, Adjacent Edge Blunder
1. a2 b3
2. a3+ a1
3. b2#

a b c
3 X O
2 X X
1 O

Edge Game, Opposite Corner Blunder
1. a2 c1
2. a1+ a3+
3. b2#

a b c
3 O
2 X X
1 X O

On the other hand, the center, opposing edge, and adjacent corners are all solid defensive options, with good chances for O if X makes a mistake.

Edge Game, Center Defense
1. a2 b2
2. b3 a3+
3. a1+ a3+
4. c1+ b1 and draw

a b c
3 O X O
2 X O
1 X O X

Edge Game, Symmetrical Defense
1. a2 c2
2. c1 a1 and draw

a b c
3
2 X O
1 O X

Edge Game, Adjacent Corner Defense, Opposite Blunder
1. a2 a1
2. c2+ b2+
3. c3+ c1#

a b c
3 X
2 X O X
1 O O

There are a couple of principles that we can observe in this. The first is that the direct approach seems less successful in an n-r game. The attacks with the best chance of working involve delayed actions that produce multiple threats. Anyone can notice when you have two pieces lined up in a row against him, but it is possible to miss the point of a ‘developing’ move and respond with a play in a bad square.

The second is that when it comes to balanced games, you don’t win so much as wait for your opponent to lose. If your adversary intends to draw, and moves with the purpose of blocking you as much as possible, there’s no reason they shouldn’t be able to get what they’re after. They are in danger when they decide to play a tricky game and try to ‘steal’ a victory. Of course, the larger and more complicated the game, the less applicable this is.

On that note, stay tuned for the next part, where we discuss some more advanced n-in-a-row games – Tic-Tac-Toe on larger boards, both balanced and imbalanced games.

* * *

Considering the ubiquity of regular Tic-Tac-Toe, it is curious that few have thought to take their game to the next level. In this article, we will explore the next step in n-in-a-row play, larger boards and larger row requirements.

It is evident after a brief look that an expanded Tic-Tac-Toe game is going to require more than three-in-a-row. For a thought experiment, consider the play of Tic-Tac-Toe+1. This game is exactly like Tic-Tac-Toe, except that an extra square is added onto the edge of the board: next to any of the existing squares, either horizontally or diagonally.

If the new square is anywhere but to the diagonal of a corner square, the game becomes a forced win for X:

New Square at a4, Game 1
1. a3 b2
2. a2#

New Square at a4, Game 2
1. a3 a2
2. b2+ c1
3. b3#

New Square at b4, Game 1
1. b2 c3
2. b3#

New Square at b4, Game 2
1. b2 b3
2. a3+ c3
3. a1#

By the same measure, a greater row requirement can lead to a game that is too easily drawn, even by the standards of a game like Tic-Tac-Toe. A brief playthrough of 4-in-a-row on a 4×4 board should illustrate the point.

A Bad, Boring 4×4 Game
1. a1 c3
2. b1 b3
3. c1+ d1
4. a2 d2
5. a3+ a4+
6. c2 d4+ 7. d3 c4+
8. b4 and a draw

The best size for a board tends to be the largest one that is not proven to be a win for the first player. For 4-in-a-row games, the only suitable symmetrical board is 5×5 [1].

Given a 5×5 board and 4r, what does the opening play look like?

Although X may have motivation to try different options in Tic-Tac-Toe, in this game, there is no reason to play anything but the center. This gives him the most room to expand his attack.

At this point, O must move in a square diagonally adjacent to the center. Anything else is a forced win for X.

Edge Defense Refutation
1. c3 b3
2. b4 d2
3. d4 c4
4. b2#

a b c d e
5
4 X O X
3 O X
2 X O
1

Time and space don’t presently allow an exhaustive listing of X’s options after a corner defense, but the strategy is focused on getting one of the possible combinations which lead to 4r in short order, which include:

  • 1 three-in-a-row, with no pieces blocking either end.

  • 2 simultaneous three-in-a-rows, either or both of which may be blocked on one end.

  • 2 simultaneous two-in-a-rows, with no pieces blocking either end.

A solid strategy for this game is to play the center of the board as a regular game of Tic-Tac-Toe. If you keep this foundation in order, the threats at the edges are easy to prevent. The biggest problem a player is likely to run into is the desire to ‘cheat’ by playing a move oriented towards getting a winning threat when they should be focused on defense.

A Sample 5×5 Game
1. c3 b2
2. b4 d2
3. e2 d4?!
(Complicates the game, but probably for the worse.)
4. d3 c2!
Saves the draw.
5. a2 b3
6. a3 a4
7. d1 e3
8. c4 b5
9. c5 and draw

Another Sample 5×5 Game
1. c3 b2
2. b4 d2
3. e2 c2
4. a2 d3
5. d4 e4
6. b1 a4
7. b3 d5 and draw

It is occasionally useful, at least psychologically, to make a move that blocks only one end of a row, when you have the option to do otherwise. This seems to be halfway between playing a solid defensive move and a ‘cheat’. Whether it will prove to be sound or unsound depends on the details of the position and whether your opponent keeps track of what you’re up to, but I suspect that it is slightly incorrect and this habit can get you punished in similar games on larger boards.

A game like 5r Tic-Tac-Toe is known and appreciated around the world. It has a variety of names but is best known as the Japanese “Gomoku”.

At sufficiently large board sizes, without or sometimes even with artificial restrictions, Gomoku is known to be a forced win for the first mover; this is true on boards as small as 15×15. [2] It appears to be unknown at present what the largest size is that is proven to be a draw.

I’ve played 5r games at 7×7 and 9×9 sizes. My expectation is that the latter is drawn, but this hasn’t yet been proven. Play allows no room for error.

I’ve never focused my attention on this game in the way that I might have, for two reasons.

Firstly, the fact that it already has many expert and professional players has a chilling factor on my level of interest. I admit being opportunistic in this respect. I would rather try to break ground in a new land and be first-rate at something than to be second or third-rate in a sprawling metropolis. At any rate, I’m not sure how much value I can offer writing about a game that has already been discussed and solved and revised and re-solved multiple times (the previous content in these articles notwithstanding).

Secondly, I discovered Ultimate Tic-Tac-Toe shortly after my first 5r series, and my enthusiasm for that game further dampened my interest in doing an in-depth study of this one. It is therefore appropriate that Ultimate (or, as we’ve taken to calling it, ULT) should be the subject of the next article in this series.

[1] There are larger boards which are said to lead to interesting games. The key phrase here is “symmetrical board”. This vintage site is informative about that, and such variants generally: “Generalized Tic-tac-toe”, by Wei Ji Ma
[2] The proof text is Searching for Solutions in Games and Artificial Intelligence, a 1994 thesis by Louis Victor Allis.

 

Where Honor is Due

This is a classic gPress post, originally uploaded November 3, 2022.

Is President Biden worse than Nero?

Anyone acquainted with history might think this question is ludicrous, but it is a point of confusion in Christendom today. Many feel it is acceptable to indulge in hearing and spreading rumors about someone, or to deride them, or otherwise rail against them, provided that it is a political leader they disagree with.

Far from defending this practice, the Bible shows reasons why we should treat President Biden with respect, most of which are also true for other elected officials.

1) He is a ruler of our country.
Although there seems to be a trend not to recognize anyone you have not voted for, the Biblical perspective is that if someone has authority, they have been ‘set up’ in that position by God. This setting up is not limited to means that we personally approve of; whether the president was ‘duly elected’ seems moot when we recall that several Biblical rulers who were acknowledged as legitimate came to power through conquest or assassination!

1 Peter 2 says to ‘honor the king’. This was likely written in reference to the Roman emperor. It is unqualified, and in context, the statement to respect someone who tradition tells us was responsible for leading a great persecution and executing Peter himself removes from us any kind of excuse. If we find this difficult to practice, it is more difficult to argue that the president is responsible for more evil-doing than either the Roman emperors generally or Nero in particular, and that his rule is an extenuating circumstance.

If nothing else, the Biblical mandate to pray for our rulers, and the spirit of that mandate, opposes the attitude behind this manner of speech.

2) He is aged.
Many use this as a jumping-off point for further dishonoring the president. However, the Bible states in Leviticus 19 that the elderly are due respect, without qualification:

“Thou shalt rise up before the hoary head, and honour the face of the old man, and fear thy God: I am the LORD.”

3) He is a person.
One of the best reasons to refrain from these kinds of comments about the president is that it is wrong to habitually talk about anyone in this manner, whatever their status in life.

It is one thing to criticize the administration’s policies, or to note when the president has done something wrong in his office as a political leader, whether it is immoral or merely imprudent. Even in this case, I would err on the side of silence, in the sense that dwelling on someone’s faults is not expedient, but I acknowledge there is an occasional need for discussion of the conduct of our government.

However, most conversation about political leaders consists of emotionally charged rhetoric, abusive language, and personal attacks. If you would not put up with hearing someone speak this way about you – or your family member, or your friend, or acquaintance – why should a stranger be any different, whatever their position? They are all humans made in God’s image.

I’m reluctant to write it out, but this point must be made. I’ve heard someone in the past thoughtlessly say that they wished for the president’s death. I wonder how many times we have entertained such ideas without understanding what is meant by it. If you say this, what you mean (intentionally or not) is that your disapproval of the president is so great that you, believing him to be a sinner, prefer him to immediately face the consequences of those sins, rather than repent. I hardly believe that Christians want this when they speak about someone, but if they don’t, they should pick a new manner of speaking.

Ultimately the president is a man like any other man. If we claim to be concerned about the salvation of the lost, the president and other political leaders are in as much a need of it as anyone else. We should not harshly reject and condemn someone whom God has not.

4) The speech is un-Christian in any case.
“Oh, so you’re a Christian? I never would have guessed.”
It is an understatement to note that someone who seems habitually angry is not modeling Christ. Reviling is no better than profanity or obscenities; the three often go together. Moreover, people judge you not only by how you treat them, but by how you treat others.

Although love, grace and peace are misunderstood nowadays, people can recognize that something is amiss when someone claims to possess these traits and yet is constantly offensive (in both senses of the word). If you can persuade a stranger that you hate a third person, you are well on your way to convincing them that you can’t love them either.

In conclusion, according to Scripture, the president of our country is due our respect, as are other public officials. You do not have to agree with or support someone’s decisions to extend them Christian love, much less the common courtesies that are ordinarily extended between strangers.

While the media wants to make us feel we are ‘familiar’ with these people we have never met, we cannot in good conscience let that supposed familiarity breed bitterness and contempt. If we do, we will have to account for it someday.

Choosing a Popular OS

This is a classic gPress post, originally uploaded April 30, 2022.

There are three mainstream families of operating systems (“OSes”) for regular users in modern computing: macOS, Windows, and Linux (which includes Google’s Chrome OS).

Each platform has pros and cons. I hope to inform less experienced users’ decisions. If you build software from source, this post is not addressed to you.

Table of Contents
macOS
Windows
The Linux family
• Chrome OS
• Debian
Conclusion

macOS

Your choice regarding macOS is made when you buy your computer. It runs on and is designed and licensed for Apple hardware, and since the chief advantage of a Mac is the operating system, it makes little sense to buy one and then run another OS full-time.

Macs are marketed as being easier to use than their alternatives. An influential IBM study found that Mac corporate users were more productive, more satisfied, and saved money compared to Windows users. Apple’s customer loyalty is recognized in the computing world. Modern Macs offer the advantage of being part of the so-called ‘Apple ecosystem’, offering superior synchronization with iPhones and other iDevices.

There are two downsides to the Mac as a platform. The first is that Apple has control over your system. I don’t mean this in the Orwellian sense, but in that your ability to customize or upgrade your machine is limited. If you don’t like the way it comes out of the box, or if something gets changed in an update, you’re stuck with it. If you have a problem, you’re left to Apple’s mercies to fix it. If it works fine, then sooner or later Apple will drop support for it, with the expectation that you will immediately replace it with a newer model.

Macs also tend to have a hefty up-front price, although the least expensive Mac you can get directly from Apple as of this writing (6/15/2025) is $509 [1], not including shipping. Used Macs retain their value quite well. You can save money in the long run, but the entry cost may be prohibitive.

Windows

Windows has been the market share leader in desktop operating systems since the mid-90s, and has usually been ‘good enough’ for most people.

Microsoft has a record of being concerned about legacy applications [2]. You can still run some early-90s programs on modern Windows PCs, which stands in stark contrast to Apple’s regular forced abandonment of old software.

If it is not the most customizable system, it is still very much so. If it is not the most stable operating system, it is more stable than it is given credit for, and harder for users to mess up than a system that assumes competence. If it is not the most intuitive system, it’s intuitive enough; the chances are that after 25 years, you already know your way around it.

There is something to be said for being consistently no less than second or third-best, but Windows is also first-rate in several areas. Windows has the largest community and support base of any desktop operating system. It is the #1 platform for computer gamers, with a catalog larger, better-supported, and better-performant than the alternatives. “Quantity has a quality of its own”, but with ready access to cutting-edge graphics hardware, you don’t have to choose between the two. Driver support is as good as it can get on the PC’s ‘open architecture’. Generally, for any third-party software that you can name, Windows receives top development priority. These are all perks.

The problem with Windows is Microsoft. Their monopolistic practices have survived to this present day, and they have an atrocious reputation for user privacy and security, underscored by their recent introduction of action-tracking AI onto users’ computers. They also have picked up the habit of spontaneously making changes that no users asked for, sometimes to benefit themselves, other times apparently for no reason but to see things change.

The Linux family

There are dozens of operating systems, but Linux distributions are the only ones that regular users would find easy to take full advantage of. I’m going to select and recommend two of them here, a corporate distribution and a free one.

Chrome OS

Chrome OS is a simple and lightweight Gentoo Linux-based rival to macOS and Windows. Chromebooks have conquered the educational market, which used to be key Macintosh territory, because they are both easy to use and inexpensive. In addition to the standard ‘web applications’ most people are used to, new Chrome OS devices offer passable support for standard Linux programs and Android apps.

There are minor problems with Chrome OS, but its ultimate issue is that it is from Google. It emphasizes cloud storage because Google is a cloud storage provider. You cannot expect privacy because Google is an advertising and search company. Whenever your personal interests and Google’s interests diverge, you have no reason to doubt that theirs will win out. Google seems more competent than Microsoft in their implementation of software, but that is not reassuring if you believe that they have nefarious intent.

Although true Chrome OS installations are built into the eponymous ‘Chromebooks’ (or Chromeboxes/Chromebases), you can deploy a Chrome-like operating system to your own Mac or PC via Chrome OS Flex. Some hardware is better-supported than others; check their list to see if your machine is certified.

Debian

In the past, I wouldn’t have peddled Debian to new users. But after applying my listed standards to recommendations, I’m short on alternatives. Besides, my continuing experience with modern Debian suggests that it has become accessible, when you are willing to learn, or can get help.

As the upstream source for most of the Linux world, Debian has an established community and has stood the test of time. It comes with little inherent bloat, is both configurable and well-supported, and should be reliable for the foreseeable future.

Depending on your machine, you might find hiccups with driver support during initial setup. For example, older Macs are known for requiring wifi drivers that need to be installed separately. Check in advance to see if your hardware has any special needs. If your machine does have trouble, someone has likely already walked someone else through a similar difficulty on a discussion forum somewhere.

Conclusion

Now that I’ve evaluated the traits of each of these platforms, I will give synopses of why you might use them.

If you have money, and either 1) own an iPhone, 2) don’t play many video games, and/or 3) view your computer as an appliance, get a Mac.

If software compatibility is important to you, including but not limited to gaming, you might go with Windows.

If you have a low budget, don’t have extensive processing needs, and/or don’t despise Google, try a Chrome device. (If you have an Android phone, you don’t despise Google.)

If you value customization, and freedom from ‘big tech’ and artificial end-of-life dates, decide for Debian.

If you have questions or comments to make about this list, feel free to drop a message by my address!

[1] This price was for a refurbished M4 Mac Mini. In Sept 2023, a M1 Mac Mini was $469, and a new low-end M2 was $599. In 2022 they were $589/$699 respectively.

[2] Windows 11 breaks some things and makes computers obsolete, but I suspect it is still better than alternatives.

The Second-Person Plural Pronoun

This is a classic gPress post, originally uploaded April 4, 2022.

The shame of modern English’s lack of a second-person plural is that the damage is self-inflicted.

Basic pronouns, modern English

Singular Plural
First I/me/my/mine We/us/our/ours
Second You/you/your/yours N/A
Third He/she, him/her, his/her, his/hers They/them/their/theirs

You can address or refer to anyone using the pronouns on the above table, unless you are directly addressing a group. In that case, you are forced to either:

1) Say only “you” and hope people infer that you meant the whole group
2) Use a lengthy combination of words: “you all”, “all of you people”, “you guys”, “ladies and gentlemen”, “to the group of human beings I am addressing”
3) Invent a new word: “y’all”, “youse”, “yinz”, “youins”

There’s an ongoing debate about which of these solutions is best. What is little-known is that we have had a solution to this all along, and the cause of our problem is cultural niceties.

Before discussing that, I feel it is helpful and educational to bring up the basic system of Spanish pronouns. This is what helped me to understand the history and development of our English language, by the parallel to another well-known language. The list will not be complete, but it will be enough to give a brief overview.

Basic pronoun forms, modern Spanish

Singular Plural
First Yo Nosotros
Second Tú, usted Vosotros (in Spain only), ustedes
Third Él/ella Los/las

At a glance, you can see that Spanish has all of the major fields covered, and then some. Why the extra ‘usted’ form, though?

The short answer is, words have connotations beyond their grammatical usage. These forms are either ‘familiar’ or ‘formal’. ‘Usted’ is the form you use when addressing someone who is socially superior, or perhaps someone else you don’t want to risk offending. (The corresponding English practice these days is to generously sprinkle ‘sir’, ‘ma’am’, or ‘miss’ through your sentences.)

The problem with having ‘formal’ and ‘familiar’ forms is the duality; if ‘formal’ language indicates respect and an acknowledgement of social standing, what does using the ‘familiar’ form imply?

Back when social statuses were clearly delineated, this wasn’t a problem, but as the world became democratized, using usted/ustedes became the practice in most of the Spanish-speaking world. Only Spain still uses the ‘vosotros’ form, and this distinction, speaking “el español del rey”, would cause you to stand out in a Latino crowd.

With this foundation laid, let’s have another look at English, or rather, English as it was.

Basic pronouns, Elizabethan English

Singular Plural
First I/me/my/mine We/us/our/ours
Second Thou/thee/thy/thine Ye/you/your/yours
Third He/she, him/her, his/her, his/hers They/them/their/theirs

These should be familiar to anyone who has read either the King James Version Bible or Shakespeare. Because those sources are also likely to be the only place they are seen, they are misunderstood, to the point that ‘thees and thous’ have become a byword for things which are considered complicated, archaic, and ‘unnecessary’.

People know the ‘thou’ form because it is a pronoun used to address God in the KJV, but they don’t know why it is used. The reason is shown in the above chart: it is the second-person singular pronoun, and God is a single person, so when someone is directly addressing God, correct Elizabethan grammar demands that form. Meanwhile, the ye form is like the modern-day ‘y’all’.

We lost this system for the same reason that the Spanish language was transformed: the forms acquired connotations of respect or disrespect, and people switched to using the ‘formal’ form all the time. English doesn’t have a dedicated formal form, so ‘you’ was made to do double-duty, as either the second-person plural or the formal second-person singular. I suspect the logic behind that choice was like that of the ‘royal we’.

Incidentally, the King James’ English is said to have been archaic even for its time. When the translators were putting together the work, the thees and thous were on their way out; they chose to stick with those words on the grounds that it was the most direct translation of the original pronouns (which it was, and still is). This choice was said to get criticism from readers who felt that the use of the familiar ‘thee’ and ‘thou’ disrespected God.

At any rate, people forgot over time that ‘you’ was supposed to be a second-person plural, which meant that we ceased to have one. But there are inevitably cases where that form of grammar is helpful, which has since become obvious. Thus, we now have a plethora of conjunctions designed to address this need, as alluded to at the beginning of this post.

The trouble is that they all have the same problem as what got ‘thou’ canned in the first place: they’re perceived to be informal!

The more things change, the more they stay the same…

On Means of Organizing Web Discussions

Suppose you have a room filled with loose folders. There are too many of them for one person to manage in a reasonable time, but you feel that you need to organize them somehow. Your friend Hubert is there to offer advice. “Let’s organize all of the folders by date,” he says, “and put them into boxes, with the newest folders being in one box, and so on.”

You suppose that’s fine, you say (although you, not having an eidetic memory of when folders were last relevant in your life, wonder whether it wouldn’t make more sense to box them based on subject), but how should the boxes be labelled? “It couldn’t be simpler,” comes the reply. “The box with all the newest files is a 1. Then we add a number for each extra box. I guess we have 167 folders here, and we should be able to put 15 folders in a box, so that’s 11.13 boxes. We put down a 12 on the oldest box. We’re good to go.”

“Don’t we have to move the folders around every time that we want to add a new folder to the newest box?”

Hubert looks at you quizzically, adjusting his smudged glasses. “Of course. But you have to move a folder every time you update it, too. What’s the matter with that?”

“Don’t you think it might be a little hard to remember where each folder is, after it’s been rearranged a few dozen times?”

“I remember everything. But if you don’t want that, that’s fine. We can just do an ascending sort. Put all the oldest folders, in order, in box one. Now the newest folders will be in the box with the largest number. Does that suit you?”

“Not really! I don’t know anything about the boxes, other than that the lower numbers are old and the higher numbers are new. I don’t know how old they are or the subject of the topics… without looking at the folders inside one, I don’t have anything. And if I do remember where a few folders are, once something happens and they get updated, everything moves and I don’t even have that. These labels are useless.”

***

This is the essence of the difficulty with most online paging systems. Nevertheless, the system was taken for granted as early as webboards of the 1990s, and it must be older than that. I would guess that this basic concept was in use on BBSes, and probably before.

Why? As someone who implemented it myself, I speculate there are three plausible reasons.

1) It is pretty easy to program (even though it is also easy to get wrong, more on that in a minute).

2) It helps keep pages a uniform length. A short page may not be a problem, but 20+ items quickly become difficult for humans to read.

3) It avoids the possibility of running into a blank page: for instance, if you organized data by time periods, it may not be obvious in advance when you pick a month whether it will be empty or not. When a page necessarily consists of a known number of existing items, that problem is avoided.

So, in spite of my misgivings, I intended to use pages for organizing threads, but ran into a problem.

***

“Let’s try this out,” Hubert says. “Please select an offset that you wish your folder selection to begin with.”

“Let’s go with 150, I guess.”

“Excellent. Processing…” Reaching for the first box, Hubert begins to count off each folder inside. You unthinkingly count with him, until he finishes and reaches for the second box. Then you come to yourself.

“What in the world are you doing? These folders don’t have anything to do with the search.”

“On the contrary, they are essential,” Hubert explains, not turning around. “We can’t know that we are selecting folders from 150 on unless we have counted out 150 folders before that.”

“You’re telling me that if we had a million folders, and I asked you to start half-way through, you would count off five hundred thousand, just so you could give me 15 folders after that?”

“That’s correct,” acknowledges Hubert, without a hint of consternation.

You sigh. “We are going to have to rethink this filing system.”

***

Many common guides on how to use a database will mention the OFFSET keyword, giving you the impression that it is the solution to all of your paging problems. What they neglect to mention is that the process works just as described in the story. This inefficiency might be tolerable if there are never that many things to search for, but if you’re assuming that nobody is going to use your system, why make it in the first place?

There are means of getting around this, some of which are more or less suitable depending on the context.

***

“So we keep track of the location of the last file in the group, and then future searches can start after that point,” Hubert explains. “Now, please enter your query.”

“Please show me the first 15 files after 150.”

Hubert shakes his head. “This system doesn’t work that way. You have to ask for the first box first. Then, if you wish to look at the second box, and so on, I can oblige you.”

“So you still can’t get to 150 without counting off every file first?”

Hubert blinks owlishly. “Not really, but this is practically irrelevant. If those files were important to you, they would be earlier in the count, not later. The later the files are in the search, the less relevant they must be.”

“What if I wanted to read older files first?”

“You would specify that in your original search, so that the first box consists of the oldest files, and so on.”

“What about the files in the middle? It sounds like they’re difficult to get to either way.”

Hubert shrugs. “There are inefficiencies in any system.”

***

This system, which some call ‘cursor-based’ and is the primary method described in a MariaDB guide on pagination optimization, is fine in certain contexts. It is the essential method search engines use (in a context where the first results are assumed to be the most relevant). But as a means of pagination on a forum or a messaging system, it seemed lacking. The premise of a forum, as I see it, is that information is being organized, not just dumped, as it might be on postmodern social media.

I had already implemented what I called ‘chapters’ for thread pagination in my messaging system (the code for which is accessible in my archive of Middling Works). The fundamental idea of chapters is that organization comes from more manual, rather than automated, processes. So, users manually manage pages by defining the start and end points of each page. This is great for managing discussions (which may have several topic shifts, which can be split into different pages), forum games (which are likely to have ’rounds’, which can be split into chapters), and fiction writing (which naturally lend themselves to being divvied up, one thread chapter for each story chapter and its commentary). But although I felt this system was brilliant for organizing posts inside threads, using it for threads has problems.

The first issue was that I was designing this system for a private messenger. So each user would see different threads, and there would be a lot of database overhead in storing each and every user’s personal arrangements for their threads.

The second issue is that these kinds of chapters break sorting systems. I wanted to be able to organize threads by the times they were updated, as well as the times they were initially created, but chapters, as I conceived them in threads, assume an ascending sort by creation time.

Finally, even if the previous points were glossed over, there’s a problem in that users might not be able to distinguish a clear pattern in the types of threads that were being made (or last updated) in certain time periods. It might be obvious in hindsight that an era was more peaceful or more tumultuous than another, but then again, it might not. In the case of longstanding threads that are accessed across multiple generations of users, you might also have enough crossover that this kind of delineation is unhelpful.

So, I settled for organizing these threads by month and year, and patched up the ’empty month’ problem by employing a conditional search: if you struck an empty month, the system would show you the next older (and next newer) months that did have content in them, so you could easily access those.

There are some practical limitations that can come up here, although they weren’t likely to in my original context. My study of forums suggests that organizing a board ‘by month’ can lead to ponderously long lists in the case of something like a support board, which can have dozens or hundreds of queries in a month.

My inclination was to fall back on an ordinary paging system in this case, and that might be the best solution for a general discussion board. When it comes to support boards, it might be wiser to encourage a system where users navigate questions by tags or keywords rather than spatial organization.

* * *

You decide to put dates on the boxes.

“It’s a viable system, if you want that kind of thing,” Hubert acknowledges. “Now, how are you going to set up the file content search?”

“I think I will leave that problem to someone with a lot of free time.”

On God’s Will – An Exploration

Let’s follow some logical series of steps. Each letter is a premise, building up to a conclusion, the bolded number. Remember, in logic, if the statements are logically valid, and the premises are true, then the conclusions must also be true.

a. Something is either God’s will or it is not.
b. God does not will things that are mutually exclusive.
c. Someone can only be doing one thing at any given point in time.
1. So, God only wills one thing at any given point in time.

This conclusion, which is key to the rest of this line of reasoning, has difficulty answering the intuitive possibility that God can want certain (say, two) things to be done within a certain time range, and isn’t particular about their order.

We can strengthen this conclusion by adding another premise: “Given two options, God is never undecided; He always has a preference.” However, there are some hypothetical questions where even that seems difficult to defend.

Suppose the following example:
It is God’s will that a peanut butter and grape jelly sandwich be made.

Is it God’s will that the peanut butter be put on one slice of bread first, and jelly on the other, and the second slice put on the first? Or the other way around? Or should the peanut butter be put on one slice, then the jelly on top of that, and then the remaining plain slice?

One can argue that one way is more rational than the other, perhaps because of risk of condiment spillage, but it is intuitively odd to say one is sinful. Something being unintuitive isn’t sufficient reason to reject it, but God has presumably given us these kinds of feelings for some purpose, so it doesn’t follow that they should be ignored, either.

While this example seems absurd, if we accept that God is interested in everything we do, it is as valid a subject for a discussion on God’s will as something apparently weightier would be.

Suppose we use a similar case where it’s even less clear what a ‘logical’ choice might be:
It is God’s will that clothes be taken out of the washing machine and be put in the dryer.

What should go in the dryer first, the shirt or the socks?

They’re all about to be dried! Who cares? Well, if we accept that God’s will is always so specific, not just about the events but the timing, we’re saying that God does care, and if God cares, that suggests that the shirt first versus the socks first is not just a problem regarding a hypothetical optimal laundering technique, but a moral issue, as the next two arguments will show.

*

a. If something is not God’s will, it is self-will.
b. Self-will is iniquity. (Matt. 7:21-24, Ezek. 28:15, Isa. 14:12-14, 1 Sam 15:23, Jer 7:24, Jer 17:9)
c. Iniquity either is sin or is the source of sin.
d. God hates sin.
2. So, if something is not God’s will, God hates it.

*

a. If God only and always wills one thing at a given point in time, anything else that might be done at that given point in time is not God’s will.
3. Thus, for any given point in time, there is only one thing that is God’s will, and any other thing that might be done at that point in time, God hates.

This is logically derived from the previous two conclusions; if they are both true, this must also be true. The difficulties and dangers resulting from this line of reasoning are beginning to take tangible form here.

*

a. Someone who does something that is not God’s will is a worker of iniquity.
b. No worker of iniquity will enter the kingdom of heaven.
4. Thus, anyone who does something at a given point in time that is not God’s will, will not enter the kingdom of heaven (provided they don’t repent).

This seems to follow deductively from conclusion 2 (and Scripture). This isn’t strictly required for the final conclusion, and it is implied in any case, but I thought it was worth spelling out, for the sake of showing the risks associated with being wrong when you have this mindset.

*

a. If God only and always wills one thing at any given point in time, then each point in time has one thing that God wills for it.
b. Someone’s earthly life consists of a consecutive series of points in time.
5. Thus, someone’s earthly life has a consecutive series of single things that God wills for it!

This follows directly from 3, and reveals the core of the problem, by suggesting people are supposed to follow an exact series of steps (or a ‘script’) in their lives.

If God reveals all those steps, they are instructions, which means that God wants robots – robots being machines that follow consecutive series of instructions for their entire ‘lifespans’.

If He doesn’t reveal (some, or even one, of) those steps, it’s still His will, so by definition, He still wants people to do those things. So in this case, they’re left to guess what God’s will is. Given the multitude of possibilities, they will probably fail. In fact, the argument can be made that even if they happened to correctly guess what God had wanted, they would still have failed, because they made their own decision about it.

In either case, someone who doesn’t follow God’s will exactly is a worker of iniquity (based on point 4).

Neither of these scenarios (wanting robots, or setting people up to fail) seem to be a positive portrayal for God.

The latter can only be sidestepped if we assume that every moment that God has not revealed His will for is a time that God doesn’t intend for the person to do anything, but rather wait until He does reveal His will.

* * *

Perhaps the strongest argument against this way of viewing God’s will is practical experience, as revealed in both Scripture and our lives.

When understanding God, it is helpful to look at God’s original plans and designs. Whether people were compliant or not, those things nonetheless were what He intended. Since God doesn’t change, this is presumably still His ideal, and thus what mankind will be returning to at the end of time (provided that He hasn’t clarified, like with human marriages, that things will be different).

In Genesis, God gives a few commands:

1. And God blessed them, and God said unto them, Be fruitful, and multiply, and replenish the earth, and subdue it: and have dominion over the fish of the sea, and over the fowl of the air, and over every living thing that moveth upon the earth.

29 And God said, Behold, I have given you every herb bearing seed, which is upon the face of all the earth, and every tree, in the which is the fruit of a tree yielding seed; to you it shall be for meat. 30 And to every beast of the earth, and to every fowl of the air, and to every thing that creepeth upon the earth, wherein there is life, I have given every green herb for meat: and it was so.

2. And the Lord God commanded the man, saying, Of every tree of the garden thou mayest freely eat: 17 but of the tree of the knowledge of good and evil, thou shalt not eat of it: for in the day that thou eatest thereof thou shalt surely die.

So:
1. Be fruitful and multiply.
2. Replenish (meaning “fill”) the earth, subdue it, and have dominion over every living thing.
3. Eat of every herb bearing seed, and of the fruit of every tree, except for the tree of the knowledge of good and evil. (If ‘green herbs’ are different from the ‘herbs bearing seed’, those would also presumably be forbidden, being reserved for animals, but it’s fair to assume this isn’t a temptation that would afflict man in their original state. Even people today don’t generally have a problem with not eating grass.)

There’s no indication that there were any more specific commands than that. As a commentator pointed out, Adam named the animals; it doesn’t say that he learned the names that God already had for them.

One of the gravest difficulties associated with this theory of will is that it denies the proposal that man should make use of the creative impulse. He and she inherit it by birthright, being made in God’s image. But if God always has specific steps to follow for every moment of someone’s life, then they are called to suppress this part of their nature, now and forever, because a being that only acts on someone else’s specific instructions cannot be truly creative. Including such a major feature in man’s makeup and never intending for it to be used seems cruel in a way that is incompatible with God’s revealed nature. Just as importantly, it is incompatible with what Genesis actually said happened. We are forced to conclude that in paradise, at least, God’s will is not always so specific.

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It is reasonable to suppose that God continued to interact with humanity closely after the fall. Cain wasn’t surprised when God spoke to him, and after his exile, the Scripture notes that Cain ‘went out from the presence of the LORD’. This suggests that the others, like Adam, Eve, and their ‘other brothers and sisters’ remained there. It’s possible that God did give direct instructions to them, but this is unrecorded until Noah. God gives Noah requirements for the Ark, but as a set of plans, it leaves a bit unaddressed. Either the Genesis account is not a complete rundown of God’s instructions, or there were some details that God didn’t speak about at the time.

Sometime later, the practical experience of the patriarchs and matriarchs, and their contemporaries, seems to be that they didn’t receive specific information for very much of their daily lives. The norm was to receive a vision or a dream by night: Job’s friend Eliphaz testified of a vision that he had some time before. Elihu was familiar with the practice and said that God often spoke in this manner and was simply not being heard (though this could not in any case inform every step taken during the day). Abimelech king of Gerar talked with God in a vision. Abraham and Jacob received periodic instructions, sometimes years or decades apart, concerning major life decisions. Laban received a warning ‘by night’. Isaac has only two recorded direct communications from God (other than his being with Abraham on Mt. Moriah), the first in which he is told not to go to Egypt, the second where he is simply affirmed in the covenant blessing. Joseph presumably had more specific interactions with God when he gave interpretations for dreams, but it isn’t clear that he was so closely directed in every act of his life. Dreams continued to be the norm in non-Jewish societies for centuries later, and were of great interest to their recipients, people as varied as Nebuchadnezzar and Pilate’s wife.

The first person recorded who definitely received ongoing, specific directions for a great quantity of things is Moses, and it is likely he received instructions on other things not recorded. Joshua’s early campaign was also closely planned out, with one notable blunder, the incident with Gibeon, on one of the occasions they did not explicitly get God’s guidance. However, in at least one of the later battles (Josh 10), Joshua receives merely an assurance of victory, with no specific details about how it is to be done. Given that those details were given in similar earlier passages, it’s possible that Joshua simply did not receive detailed instructions in the later battles.

This sets the pattern for most of the rest of the Old Testament. Other than receiving revelation in dreams, there were two common means of inquiry in Israel: by the priest, or by a prophet.

In the first case, the leading priest with the high priest’s equipment somehow inquired by the special Urim and Thummim. The norm seems to have been that the questioner asked God a specific question and got a succinct response, usually either a yes/no, or a selection from a list of options (Judges 20:18). David’s inquiry about the Ziphites in 1 Sam. 23 seems indicative of this means. At one point, he asks a complex question and only receives an answer to the second part of it, requiring him to ask the first part again, showing limitations in the system. On the bright side, when the priest was available and no sin had been committed (1 Sam 14:37), this seemed a reliable way to get direction ‘on demand’ when the questioner needed it, good for urgent situations. This was likely primarily used by the king, or other important state officials, for national problems. By Ezra’s time, there were no priests available who could use this means, and history indicates that there never were again.

In the second case, one could inquire of a prophet; provided that they were a true prophet and had inspiration, they could give you an answer. This would typically be lengthier and more conversational, and provide details, instructions, and explanations that were lacking from the priestly answers. The only ‘downsides’, as it were, were that it depended on such a prophet being available (which wasn’t always a given), and that the messages didn’t always arrive in a predictable fashion. While there are cases where the prophet responded or prophesied instantly, sometimes those statements were prepared by God in advance (such as Ahijah replying to Jeroboam’s wife). There are other times when it only came during the night, or after some kind of event. When David told Nathan about his desire to build a temple, Nathan didn’t give him an immediate prophetic response, but rather his personal opinion that it sounded like a great idea; he only got the word from God that said otherwise that night, and gave it to David the next day.

The prophets were likely more accessible for most than the priest. In Samuel’s time it was not unheard of to inquire of the ‘seer’ for minor or domestic problems, like missing livestock.

Sometime after Malachi, in accordance with Amos 8:11, there were no active prophets in Yehud/Judaea, especially not any who left a written record. An apocryphal book of Maccabees states outright that there were none in their day, indicating that this was common knowledge. (However, this apparently does not mean that there were no people who heard from God for hundreds of years, or that there were absolutely never any more prophets until John, because Luke states that at the time Jesus was born, Simeon had heard from the Holy Spirit that he would not die without seeing the Christ, and also that Anna was a prophetess.)

Finally, it was also not unheard of to cast lots, a process similar to rolling a die or drawing a paper out of a bag, to determine a course of action. The Proverbs seem to speak on it favorably (in 16:33, and 18:18), and note that its decision ‘comes from the LORD’. The process seemed primarily used in land or position distribution cases where one person’s claim was no better than another’s, although Saul apparently resorted to it when a priestly inquiry failed. The practice also found acceptance among heathens, as shown in Jonah, where the sailors cast lots to determine who was responsible for their plight. There’s no evidence that this was ever a major, much less preferred, means of settling a matter or finding God’s will.

In conclusion to this section about the Old Testament period, it is fair to say that most people during this time did not get regular ongoing instructions from God, and even when they did inquire by the means available, they didn’t always receive exhaustive answers.

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How did things happen in the New Testament?

Given that Jesus was fully God and fully man, it is difficult to understand the exact interaction between His dual natures. We know that He lived a sinless life, and was astounding teachers of the law at a young age, so it is surprising to find that He somehow ‘grew in wisdom’, according to Luke. Some knowledge was apparently not made available to Jesus in the earthly sense, as He notes that nobody, not even the Son, knows the time of His return but the Father. But in any case, He commented that He ‘only did what He sees the Father doing’, and that the ‘Father shows Him all He does’.

When Jesus was physically there, He apparently did not give the disciples continuous instructions in the sense of telling them what to do every moment of the day. There were a number of cases where they did receive clear instructions, but more often than not, Jesus seemed more interested in having a dialogue and asking questions to show them the state of their hearts and minds. He does mention that at some point in the future, when they are being prosecuted and persecuted, they should take no thought to what they will say, because the Spirit, not them, would speak through them. He states at one point that after He leaves, the ‘Comforter’ will come, teach them ‘all things’, and ‘bring all things to remembrance’ (in that case, the things Jesus had already said to them). While Jesus spent forty days with them after the resurrection, surprisingly little of it that is recorded is in new instructions.

After the Ascension and before Pentecost, the apostles felt the need to replace Judas with another man who had been with them from the beginning, and couldn’t see anything qualifying one of the candidates over the other. In this case, they resorted to the unusual expedient of casting lots. This apparently resolved their problem, but it’s worth noting there’s no record of any believer doing so again afterwards.

Visions and dreams were still a common means of learning God’s will, agreeing with Joel’s prophecy. Cornelius received a vision of an angel telling him to send men to bring Simon Peter. Peter, before Cornelius’ men came to the house he was staying, he received a certain vision, and apparently reacted the same way, three times. Peter also initially mistook his deliverance from prison for a night vision. Saul encountered Christ in a vision on the way to Damascus. Some years later, after having attempted to go to various places and having been somehow forbidden by the Spirit (by what means is not clear), he had a vision of a Macedonian man which inspired him to take the team to Macedonia. Finally, perhaps the best-known New Testament vision was when John received an intense vision revealing Christ as king, along with many end-time events.

Angels sometimes made appearances, whether in visions or not. Zechariah, Mary, and Joseph all received messages from angels (albeit this was before the church period). Philip received instruction from ‘the angel of the Lord’ telling him to go to the desert (before he met the Ethiopian eunuch). The angel of God appeared to Paul one night while he was on a ship to Rome, giving an assurance that only the ship would be lost. Paul called down a curse on anyone, even ‘an angel from heaven’, who would teach a gospel other than what the church had already been taught.

After Pentecost, although there are a number of cases where God, or the Spirit, speaks, or prevents someone from going somewhere, there’s no indication that the majority of believers received continual instructions from God for their daily lives, in the sense of having every moment spoken for. The need for a council for the decision about whether Gentile converts should be Judaized, and the disputes and mistakes of various apostles seem to prove this point. The fact that Barnabas and Paul could not agree on something, and that this major argument (rather than a mutual agreement) caused them to part ways, indicates that at least one of them, if not both, were not receiving specific guidance for each step of the process in that situation. If the ‘greats’ of the book of Acts, who wrote the New Testament, and in many cases personally saw Jesus, didn’t continually experience this, it seems unreasonable to expect to do much better today. And given the great quantity of explicit teaching, it seems that a major focus of the New Testament is on transformation of the mind and heart with doctrine and principles, not only or even primarily with step-by-step instructions. If this could be true back then, and the believers be in God’s will then, it seems reasonable to expect, given that it’s the same church, that things can’t be different now.

* * *

This exploration caused me difficulty. The logic was there, and had yet to be mathematically refuted, and yet, practical experience (in Scripture, no less) strongly indicated that the conclusions can’t be true. Given the choice between the two, I had to conclude that there was something wrong with the logic, but what that was remained to be seen.

Out of curiosity, I asked Google Gemini to evaluate the logic of these statements, and it hit on a point which I had never noticed before: the weakness of c1, in its assumption that a person can only do ‘one thing’ at a time.

This was based on my understanding gained from scientific reports suggesting that people can only fully concentrate on one thing at a time, so that there is no such thing as true ‘multitasking’. When we appear to be doing several things at once, we are really just rapidly switching between tasks. I was also perhaps influenced by the play of a chess game, in which players make a series of single decisions.

But even if this pattern describes human behavior on some level, it doesn’t do so in a manner that is sufficient to back up the conclusions. We often do two things at once: we can type and think about what we’re about to type at the same time, or we can make a sandwich while contemplating what we’re going to have for dinner. The point can be made that the thought is ‘conscious’ and that the action is more ‘automatic’, but do we claim, then, that the physical activity, taking action on a previous thought, does not count as a ‘thing’ that we are doing? That seems absurd.

We can attempt to salvage it by combining the two, so that ‘one thing’ refers to the totality of what is being performed (all thoughts and actions, collectively), but by doing so, we butcher semantics. Whatever ‘one thing’ may be, it is a crime against language to use it, without qualification or explanation, to refer to a group of actions. If we had meant a group, we should have said so in the first place. But if we did that, we would be allowing that there might be several actions that are not mutually exclusive, and then the argument would be defeated before the first conclusion.

While we still may not know, from this exploration, what God’s will is, we can at least come away knowing something about what it is not: a great mystery that must doom human attempts to follow it.