IBM Ponder This #068 · December 2003

This month's puzzle is based on a suggestion from R. Nandakumar.

We are interested in the possibility that a given polygon P can be dissected into some finite set of (at least two) mutually similar polygons qi, which we will call "tiles". The tiles qi are similar to each other (each pair qi,qj is related by scaling, rotation, and/or translation), but not necessarily similar to the original P. We will call this a "similar dissection".

Example 1: if P is any triangle, we can dissect it into four smaller triangles qi, each similar to P and congruent to each other, by joining the midpoints of the sides of P.

Example 2: a parallelogram P can be dissected into two triangles by cutting along a diagonal. Further dissect one of those into four smaller triangles. Now the five triangles are similar to each other (but not all congruent), and not similar to P.

Puzzle (Part 1): Show that every trapezoid (quadrilateral ABCD with sides AB and DC parallel, no other assumptions) has a "similar dissection".

Puzzle (Part 2): Exhibit, with proof, a quadrilateral which has no "similar dissection".

A bonus puzzle for your amusement. Please don't send us the answer, and we're not going to publish the answer.

Two players play a game. Fifty coins are in a line; their values are positive numbers known to both players. Player 1 selects and removes either the leftmost or the rightmost coin. Player 2 selects and removes either the leftmost or the rightmost coin from the remaining 49. Continue until all coins are gone. Show that Player 1 has a strategy by which he can accumulate at least half the total value of the coins.

This problem will appear in a forthcoming book Mathematical Puzzles: a Connoisseur's Collection, by Peter Winkler, published by A K Peters, available January 2004.

Solution

Best opened after a real attempt

To be added.