## December 2003

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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 q_{i}, which we will call "tiles". The tiles q_{i} are similar to each other (each pair q_{i},q_{j} 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 q_{i}, 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.

We will post the names of those who submit a correct, original solution! If you don't want your name posted then please include such a statement in your submission!

We invite visitors to our website to submit an elegant solution. Send your submission to the ponder@il.ibm.com.

*If you have any problems you think we might enjoy, please send them in. All replies should be sent to:* ponder@il.ibm.com

**Challenge:**
11/03/03 @ 01:00 PM ET

**Solution:**
11/30/03 @ 01:00 PM ET

**List Updated:**

### People who answered correctly:

**Dan Dima** (12.7.2003 @04:33:54 PM EDT)

**Hagen von Eitzen** (12.8.2003 @04:18:10 AM EDT)

**Gyozo Nagy** (12.8.2003 @10:30:14 AM EDT)

**Aditya Utturwar** (12.8.2003 @02:23:00 PM EDT)

**Kennedy** (12.9.2003 @02:21:11 AM EDT)

**Alex Wagner** (12.9.2003 @07:15:00 PM EDT)

**Michael Brand** (12.9.2003 @02:54:21 PM EDT)

**Algirdas Rascius** (12.12.2003 @12:01:00 PM EDT)

**Mark Stubbings** (12.15.2003 @08:43:00 AM EDT)

**David Friedman** (12.18.2003 @01:44:48 PM EDT)

**John Ritson** (12.22.2003 @06:13:07 PM EDT)

**IQSTAR** (12.31.2003 @03:30:44 PM EDT)

**Attention:** If your name is posted here and you wish it removed please send email to the ponder@il.ibm.com.