Each of the triangular numbers that are below 1000 is written on a separate card. Some of these cards are then placed in a circle so that, viewed from the center, the right-hand digit of each card matches the left-hand digit of the card to its right. No tricks are used such as turning a card upside-down.

What's the maximum number of digits that can appear on such a subset of the cards?

There are 44 triangular numbers below 1000.
Of these 16 either begin and/or end with 1, 3, 5, 6 or 8:

  15,  36,   55,   66, 105, 136, 153, 171,
325, 351, 378, 528, 561, 595, 666, 861.

In the instance of 8 only one number begins with it, thus there can be only one pairing for it, hence only either 378 or 528 may be used but not both.

I propose, initially at least:
153, 325, 55, 595, 528, 861, 171, 105 and 561.

This uses 9 of my selected 16 and has 26 digits.

Edited on January 10, 2008, 7:57 am

Posted by brianjn on 2008-01-09 21:23:30