John Conway, inventor of the Game of Life, died recently from covid-19 complications.

Not many people know this, but one of the vast number of things that he investigated in his life was peg-games.

I thought of him and his math when I found, in the southwestern room of Ceporah Tower on Arteum, a half-played peg-game:


These games are usually single-player games. This is an English-style board, played by filling all pieces apart from the central square. By jumping and taking pieces horizontally or vertically, the player must remove all pieces but the last, which must end up in the center - the inverse of the starting board, like so (via http://recmath.org/pegsolitaire/):


For those of you unfamiliar with nerd-sniping, here's what happened when I saw that board...



First thoughts:
- Is this half-played game a possible layout, assuming normal rules and starting positions?
- If so, is it a solvable layout or have they screwed themselves at this point?
- Where are the missing pieces?

Next, there's the pieces that have been taken. If this is the standard layout, we know that there're mathematical rules that must be followed.

    1 2 3
    2 3 1
1 2 3 1 2 3 1
2 3 1 2 3 1 2
3 1 2 3 1 2 3
   3 1 2
   1 2 3

If you number the diagonals in the board 1, 2, 3, 1, 2, 3 etc, as I show above, then we know that each move consists of moving a piece from a "1", taking a piece at a "2", and adding a piece at a "3"... or some other order of those, but for sure, adding a piece to one of them, and removing one from each of the others. So if there was an odd number of things on the "3" spots before, now there's an even number, because we either added or removed one. Every move, the oddness or evenness of these numbered groups flips: if they were "1=odd, 2=odd, 3=even", then they become "1=even, 2=even, 3=odd", and then back the next move.

So we can say on even-numbered moves, the oddness or evenness is always the same throughout a game.

In fact, we can take it a little bit further. If you subtract the number of pieces in one of our numbered groups from the number of remaining pieces, or from the number of moves made, then the parity of that result stays the same *every* move, since those are counting things, which naturally flip from odd to even every step, because that's what odd and even means. That means the oddness and evenness (the "parity") for those results is always the same, on both odd and even moves, throughout a game. If we use the number of pieces remaining, then it doesn't matter how many pieces we started with, which is probably a good idea for our case, since we're not sure about those missing three pieces.

So we can study the normal layout (every piece there except the center one) and see that we have N-1=Odd, N-2=Even, N-3=Odd parity. As a double-check, we know that the result must have either zero or two of the groups be odd, because it started with an even number of pieces, and the three groups added together must add to that even number.

And then we look at the board ingame (or in the above screenshot), and see N-1=Odd, N-2=Odd, N-3=Even parity.

These cannot be the same game! This layout is not possible from the usual start point!

Is it solvable? Well, we know it isn't, since we know it must be odd, even, odd because that's how the normal game starts, and so how it must end. But we can prove that also by saying, well, we can see that the middle spot is a 2, so there must be zero pieces in groups 1 and 3, and one piece in groups 2. Subtract those from the one piece we have left on the board, and you get 1, 0, 1: odd, even, odd. So we have proven two ways that the "single piece in the middle" cannot be the goal of the game this Psijic was playing!

Thanks, Conway!

So, if the usual start point and endpoint are not possible... what game was being played, here?

[Aside: I know! Of course it's perfectly possible that the person who made this board object just placed the pieces in a "believable looking" half-played layout. But I know that I, as a game dev, given a chance to show a half-played game, would do more than just make it look "OK". Definitely, 100% certainty, no matter what a time-crunch I was under to make it. If only because I know how people analyze half-played chess-games in media.]

I'm going to assume the rules are the same, otherwise the possibilities are infinite, and more importantly because then we can't apply Conway's logic.

Since we numbered the diagonals one way (the '/' diagonals), we can also look at the diagonals going the other way (the "\" diagonals), this time numbering them 4, 5, and 6.

    4 5 6 
    6 4 5
6 4 5 6 4 5 6
5 6 4 5 6 4 5
4 5 6 4 5 6 4
    5 6 4 
    4 5 6

With the usual starting position it's symmetrical, so as we'd expect, the parity for 4, 5, and 6 is again Odd, Even, Odd. But the layout we see ingame is Even, Even, Even. WHAT? Doesn't that mean it must start and end asymmetrically?

Yes it does, since these numbers are just a mirrored copy of the 1,2,3 numbers. In fact, I should have realized it was asymmetrical when the parities for the first diagonals were odd, odd, even. If the first and last are different, then the board must be asymmetrical as the number 1 is in the same positions as 3, mirrored along the '/'[ diagonal. So this shape must be asymmetrical about one diagonal, but not necessarily the other.

Well, what if we start by assuming that the "missing" pieces are deliberately missing?

So instead of starting with 32 pieces, they were playing with an asymmetrical arrangement of 29 pieces

29 pieces is a significant in Elder Scrolls numerology, it seems (per https://www.reddit.com/r/teslore/comments/16cbvb/numerology_in_the_lessons_pt_2/ - as is its inverse on this board, 4.

So we're looking for a starting layout that has 4 pieces missing, and has Odd, Odd, Even : Even, Even, Even parity. and can be reached from the current position on the board by playing backwards. Does such a layout even exist? Well, we know the parity must be possible, since we're looking at one. So, can we add those five removed beads back onto the board, "unplaying: the game, in order to get a layout that "looks good" and has our desired parity?

One layout that would match is with three of the four corners of the center 3x3 missing, and the center piece missing. A valid goal to play for would be to get the inverse of that: just three of the four "inner corners" and the center piece. I'm not 100% sure that's possible, but it has the same parity, and doesn't violate Conway's "Rule of Threes" (if playing to get an inverse, the starting vacancies must be in the same position as ending pieces, or a multiple of 3 spaces away).

  ###  
  ###  
####O##
###O###
##O#O##
  ###  
  ###  

The Psijic Order's symbol https://en.uesp.net/wiki/File:LG-back-Psijic.jpg IS a triangle with a circle in the center, and this arrangement is about the closest you can get to that on this board, so... maybe?

I know what you're thinking: it's almost certain that this is a random collection of beads on the board, made almost certain by the missing pieces, and the asymmetrical parity. You're probably right, but I had fun, and it was nice to be able to use Conway's work in a game other than the Game of Life.

And now this asymmetrical peg-game layout being a problem posed to Psijic acolytes is part of my head-canon, so there.