Maths Olympiad Prep

Track / Stage 6 / 134 of 400 #1134 of 1964

Problem 1134

National olympiad, first round
Combinatorics Difficulty 6.2 Prove it

Hanging a painting on 4 nails in such a way that if any one of the nails is removed, the painting falls. Is your solution "optimal" (by the way, how would you define an "optimal" solution)?

[!] If you think your solution is optimal, show it.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

An interesting measure of optimality is the number of letters required to write the word associated with the solution. As a solution, one could choose

[[[a1,a2],a3],a4]=a1a2a1−1a2−1a3a2a1a2−1a1−1a3−1a4a3a1a2a1−1a2−1a3−1a2a1a2−1a1−1a4−1 \left[\left[\left[a_{1}, a_{2}\right], a_{3}\right], a_{4}\right]=a_{1} a_{2} a_{1}^{-1} a_{2}^{-1} a_{3} a_{2} a_{1} a_{2}^{-1} a_{1}^{-1} a_{3}^{-1} a_{4} a_{3} a_{1} a_{2} a_{1}^{-1} a_{2}^{-1} a_{3}^{-1} a_{2} a_{1} a_{2}^{-1} a_{1}^{-1} a_{4}^{-1}

which contains 22 symbols, but better is

[[a1,a2],[a3,a4]]=a1a2a1−1a2−1a3a4a3−1a4−1a2a1a2−1a1−1a4a3a4−1a3−1 \left[\left[a_{1}, a_{2}\right],\left[a_{3}, a_{4}\right]\right]=a_{1} a_{2} a_{1}^{-1} a_{2}^{-1} a_{3} a_{4} a_{3}^{-1} a_{4}^{-1} a_{2} a_{1} a_{2}^{-1} a_{1}^{-1} a_{4} a_{3} a_{4}^{-1} a_{3}^{-1}

which contains only 16.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.