- C.~D.~A.~Evans, J. D. Hamkins, and N. L. Perlmutter, “A position in infinite chess with game value $\omega^4$,” to appear in Integers, vol. 17, 2017. (Newton Institute preprint ni15065)
`@ARTICLE{EvansHamkinsPerlmutter:APositionInInfiniteChessWithGameValueOmega^4, author = {C.~D.~A.~Evans and Joel David Hamkins and Norman Lewis Perlmutter}, title = {A position in infinite chess with game value $\omega^4$}, journal = {to appear in Integers}, FJOURNAL = {Integers Electronic Journal of Combinatorial Number Theory}, year = {2017}, volume = {17}, number = {}, pages = {}, eprint = {1510.08155}, archivePrefix = {arXiv}, primaryClass = {math.LO}, url = {http://jdh.hamkins.org/a-position-in-infinite-chess-with-game-value-omega-to-the-4}, month = {}, note = {Newton Institute preprint ni15065}, abstract = {}, keywords = {}, source = {}, }`

**Abstract. ** We present a position in infinite chess exhibiting an ordinal game value of $\omega^4$, thereby improving on the previously largest-known values of $\omega^3$ and $\omega^3\cdot 4$.

This is a joint work with Cory Evans and Norman Perlmutter, continuing the research program of my previous article with Evans, Transfinite game values in infinite chess, namely, the research program of finding positions in infinite chess with large transfinite ordinal game values. In the previous article, Cory and I presented a position with game value $\omega^3$. In the current paper, with Norman Perlmutter now having joined us accompanied by some outstanding ideas, we present a new position having game value $\omega^4$, breaking the previous record.

In the new position, above, the kings sit facing each other in the throne room, an uneasy détente, while white makes steady progress in the rook towers. Meanwhile, at every step black, doomed, mounts increasingly desperate bouts of long forced play using the bishop cannon battery, with bishops flying with force out of the cannons, and then each making a long series of forced-reply moves in the terminal gateways. Ultimately, white wins with value omega^4, which exceeds the previously largest known values of omega^3.

In the throne room, if either black or white places a bishop on the corresponding diagonal entryway, then checkmate is very close. A key feature is that for white to place a white-square white bishop on the diagonal marked in red, it is immediate checkmate, whereas if black places a black-square black bishop on the blue diagonal, then checkmate comes three moves later. The bishop cannon battery arrangement works because black threatens to release a bishop into the free region, and if white does not reply to those threats, then black will be three steps ahead, but otherwise, only two.

The rook towers are similar to the corresponding part of the previous $\omega^3$ position, and this is where white undertakes most of his main line progress towards checkmate. Black will move the key bishop out as far as he likes on the first move, past $n$ rook towers, and the resulting position will have value $\omega^3\cdot n$. These towers are each activated in turn, leading to a long series of play for white, interrupted at every opportunity by black causing a dramatic spectacle of forced-reply moves down in the bishop cannon battery.

At every opportunity, black mounts a long distraction down in the bishop cannon battery. Shown here is one bishop cannon. The cannonballs fire out of the cannon with force, in the sense that when each green bishop fires out, then white must reply by moving the guard pawns into place.

Upon firing, each bishop will position itself so as to attack the entrance diagonal of a long bishop gateway terminal wing. This wing is arranged so that black can make a series of forced-reply threats successively, by moving to the attack squares (marked with the blue squares). Black is threatening to exit through the gateway doorway (in brown), but white can answer the threat by moving the white bishop guards (red) into position. Thus, each bishop coming out of a cannon (with force) can position itself at a gateway terminal of length $g$, making $g$ forced-reply moves in succession. Since black can initiate firing with an arbitrarily large cannon, this means that at any moment, black can cause a forced-reply delay with game value $\omega^2$. Since the rook tower also has value $\omega^2$ by itself, the overall position has value $\omega^4=\omega^2\cdot\omega^2$.

With future developments in mind, we found that one can make a more compact arrangement of the bishop cannon battery, freeing up a quarter board for perhaps another arrangement that might lead to a higher ordinal values.

Read more about it in the article, which is available at the arxiv (pdf).

See also:

- Tranfinite game values in infinite chess
- The mate-in-$n$ problem of infinite chess is decidable
- Johan Wästlund’s MathOverflow question: Checkmate in $\omega$ moves?
- Richard Stanley’s MathOverflow question: Decidability of chess on an infinite board

I don’t get what the bishops marked blue are for. Firing one doesn’t force white to advance a guard pawn. Only green bishops do.

Also, I don’t see how the bishop cannons force a ω² delay. Each only fires up to ω bishops, each with a finite delay.

Yes, the blue bishops are not forced-reply moves, and this is important, because if black had infinitely many immediate forced-reply moves available, then the game would be a draw by infinite play, since black could simply make those moves indefinitely. Rather, for black to move the initial bishop (blue) is to open up a new cannon, and it is precisely after such a move that white gets to make progress in the main line up in the rook towers.

The bishop cannon battery arrangement is a value ω² delay, because black can count down from ω² using it with all forced reply moves (except for the initial move). (And that is essentially what value ω² means; in particular, the checkmate always occurs in finitely many moves, so we are not talking about infinitely long play with the delay). If when counting down from ω², black wants to announce ω*n+k, then he should open up the cannon of size n, and fire the first bishop to a gateway of size k. After the opening move, which is not forced reply, he now has all forced-reply moves that will enable him to count down from this ordinal however he wants. Going from ω*n+k down to ω*n in k moves inside the gateway, he then fires the next green bishop to announce value ω*(n-1)+k’, by moving it to attack gateway k’. And so on. Thus, black can simulate counting down from ω² in this way, with the reusable bishop cannon battery.

Thanks, now I’m clear on the blue bishop’s significance.

But I still don’t get the green bishop. Take a cannon with one green bishop. It can force 6 (?) moves. Two bishops could force 12 moves, three bishops 18, up to ω bishops that force 6ω moves.

Maybe each gateway could be made more and more convoluted, so that a gateway of size ω would take ω moves to get out of it. Use it with a cannon size ω and you’d get a ω² delay.

It seems to me that perhaps there may be some confusion about how game values work. A game has value $\omega$, not because it takes $\omega$ many moves for white to win, but rather because black can announce on his first move an arbitrary number $n$, and play so as to avoid checkmate for at least $n$ moves. White still wins every instance of the game in finitely many moves. If you read the introduction of the paper (click through to the arxiv for a pdf), it explains the game values in terms of such announcements. In particular: white will win in finitely many moves, and none of the plays are infinite, even though they have infinite value. By the way, each green bishop, when it is fired out of the cannon, can force any number of moves, depending on which gateway wing it attacks. So each green bishop, when played, is like an announcement that black is making right then, in terms of how much longer he will be able to play forced-reply moves before the next announcement.

Now I get it! Each bishop attacks a _single_ gateway, that is arbitrarily large, stopping once at each spot marked in blue, thus forcing every white bishop in the gateway to make a move. Thanks.

Yes, that’s right. And each bishop can attack a gateway as big as it wants, when it first moves out of the cannon.

Typo at page 14 of the pdf file on arXiv:

“and so black will advance the pawn at k3 to defend”

should be

“and so white will advance the pawn at k3 to defend”

I read both of your chess papers and was greatly amused. Thank you.

Thanks very much! I have now corrected this in the official version, which will be uploaded to the arxiv later on.

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