You are studying the effects of gravity on clay spheres. You conjecture that they will shatter... but at what height? You want to find out the smallest integral height in meters from which the clay will fall and shatter.

Unfortunately, you only have four identical clay spheres, at least until the company that makes them starts returning your calls. Also, you only have enough time for 8 tests, during which the general area will be cleared of people. Last time someone did such an experiment, an egg... well, it was messy. Up to what height can you test the effects of gravity on the clay?

We might think of this as an eight digit binary number where ones represent breaks and zeros represent bounces (or if you prefer, thuds). In that case we are looking for the number of unique eight digit binary numbers with at most four ones.

After the fourth pot breaks we will represent the remaining turns with zeros, so 11110000 will indicate that a pot will break at any height greater than or equal to one meter, 11100001 will indicate that pots break at five but not four meters while 11100000 indicates pots break at six but not five meters.

The number of unique strings of digits is the sum of 8C4, 8C3, 8C2, 8C1, and 8C0. (this is also given by 2^7 + (8C4)/2) or 163.

This means we can be sure the pot survives heights from 0 meters through 162 meters. The string 00000000 suggests the pot survived a 162 meter drop but we do not know what else above that it might survive.

Edited on May 5, 2006, 2:16 pm

Edited on May 5, 2006, 9:50 pm

Posted by Eric on 2006-05-05 02:19:37