Dan, thanks for the MSRI math problems. The sages' hats (quoted below) was about my speed. Spoilers. I liked finding that solving a little problem was one way to solve the whole problem, then realizing that sometimes the big problem *was* the little problem, so it didn't do any good to look for another method, I had to solve *that* little problem. Involutions were involved (pun). OEIS A000085 counts the number of involutions of n things. I printed out entries 0..40 and their factorizations... They seem pretty "primey"... |Basically, although the recurrence for A000085 is fairly simple, it has the talent of *ruling out* a sizeable number of prime factors (for instance 3, 7, 11, 17...). |||OEIS A264737 lists primes that do appear as factors of A000085 entries. So (except for the powers of two in every entry) you get numbers with fewer, larger prime factors than... my intuition expects anyway. | What a woild. --Steve |> Subject: [math-fun] Some math puzzles
From: Dan Asimov <dasimov@earthlink.net> Date: 5/29/20, 4:58 PM
Here are four problems from the puzzle column in the latest newsletter from MSRI, faintly edited. ... 3. Suppose there are n hats, each with a different color. These hats are placed on the heads of n sages. All of the sages know all the colors: their own hat and everyone elseÂ’s hats. A referee then announces the "correct" hat color that should be on the head of each sage. The sages are then allowed "swap" sessions: in one session disjoint pairs of sages are allowed to interchange their hats. Can the sages fully correct their hat colors in two swap sessions? -fin-