# Seven problems from Erich Friedman's *Math Magic*

Open problem settled, verified by computation. Vamshi Jandhyala.

> Three armies of five bishops fit on a 5 × 5 board and three of eight on 6 × 6, and no more: Friedman's two values.

Canonical: https://vamshij.com/research/math-magic-problems
Code: https://github.com/jvvk/mathematics/tree/main/math-magic-problems
Paper (PDF): https://github.com/jvvk/mathematics/blob/main/math-magic-problems/paper/paper.pdf

Erich Friedman's [Math Magic](https://erich-friedman.github.io/mathmagic/) poses a problem each month and keeps a [list of unsolved problems](https://erich-friedman.github.io/mathmagic/unsolved.html). As of 9 October 2026 the list shows problems 15, 21 and 24 as open, and the [March 2005 page](https://erich-friedman.github.io/mathmagic/0305.html) on armies of bishops asks for two exact values. The paper settles these four, and makes progress on problems 6, 28 and 33, which the list also shows as open:

- **Armies of bishops.** `B(3,5) = 5` and `B(3,6) = 8`: three armies of five bishops fit on a 5 × 5 board and three of eight on a 6 × 6 board, and no more. A placement is valid exactly when every diagonal carries one army, which turns the question into an exhaustive count over labellings of the diagonals.
- **Touch cycles (unsolved problem 24).** No finite arrangement of three colours of king has touch counts `(1,3,6)`, `(1,6,3)` or `(2,3,4)`, nor `(1,4,5)`. An extreme king reduces any finite arrangement to a finite formula, refuted with a DRUP certificate. Each listed triple does have a periodic arrangement of the whole plane, so finiteness is essential.
- **Capture digraphs (unsolved problem 21).** The 2-regular digraphs `q`, `r` and `s` on six points are not the capture graph of any chess position on any board.
- **Magic polyomino squares (unsolved problem 15).** The first listed heptomino can never be arranged in a square with equal row and column counts. The proof is an explicit weighting of rows and columns.
- **Slab cubes (unsolved problem 6).** A cube tiled by one `k`-slab (a `k × ik × jk` box) for each `k = 1, …, n` has side at most `n(n+1)/2`, with equality only for `n = 1, 3`; no such cube exists for `n = 4, 5, 6`. The bound is proved; the non-existence is certified by two programs, one with a constraint solver and one without.
- **Prime signatures (unsolved problem 28).** The table's entry `1022303⁴` in cell `(4,41)` belongs in cell `(4,411)`, where it is the smallest value (proved, using Ljunggren's theorem on `x² + 1 = 2y⁴`); cell `(4,41)` is empty. The two open cases have no solution below `10¹⁷`.
- **Optimal tournaments (unsolved problem 33).** Exact optima over all adaptive schedules: ranking 3 players in 7 games succeeds with probability `2992/6561` (published `2944/6561`), 4 players in 7 games with `1216/6561` (published `1136/6561`); three entries for 5 players are corrected, and every other published value checked is optimal.

Preprint v2, 9 October 2026 (v1 the same day had the first four results), not peer reviewed. Some results are computer-certified and some proved by hand; the paper says which, and for each certified result what was computed, why it suffices, and how it was checked independently. No Lean formalisation: the certified results are exhaustive counts and solver refutations, checked by certificates and by programs written independently. The author used AI tools (Claude, Anthropic; Codex, OpenAI) in this work, as described in the paper, and is responsible for its content.