There is a set of cubes of three different colors: red, blue and yellow and their edges are of integer length.

In the following statements the bold capital letters refer to specific digits with a one-to-one correspondence of digit to letter.

The volume of each red cube is NIL.
The face of each blue cube has NO area.
The volume of each yellow cube is ZERO.
The total volume of all the cubes is NOTHING.

There are NO cubes of one of the colors, and NONE of another.

How many cubes are there of the remaining color, and what is that color?

first off NIL must be a 3 digit cube, NO a 2 digit square, and ZERO a 4 digit cube.  Now each of these must also have no repeaded digit.  Thus the possible values for each are

NO: 16,25,36,49,64,81

NIL: 125,216,512,729

ZERO: 1728,2197,4096,4913,5832,6859,9261

now with a little examination we see that (16,125) and (25,216) are the only pairs where N holds the same value, but (16,125,4096) is the only set of values where N and O hold the same values throughout.  So now this is what we know that the volumes are as follows

Blue: 4^3=64

Red: 125

Yellow: 4096

now we know that 2 of the quantities are 16 and 1610 and that the total volume is 16TH21G

now 1610 can't be the quantity for Yellow cubes because then the total volume would exceed 16TH21G.  Furthermore we know that G must be either 7 or 8 because we know that the last digit of the total volume of Yellow and Blue sqaures must be even or zero and the last digit of the total volume of red squares must be 0 or 5  thus there is no combination that adds up to 3.  Thus we are left with only 4 possibilities for the total volume which are

1637218,1638217,1673218,1683217

taking each of these in turn and subtracting either 16 yellow volumes and 1610 Red volumes, 16 yellow volumes and 1610 blue volumes, 16 red volumes and 1610 blue volumes, and 16 blue volumes and 1610 red volumes.  The only combination that results in a number divisible by the remaining colors volume is

total Volume: 1637218

thus that makes T=3 H=7 and G=8

thus the remaining quantity is 21413 of blue cubes

Posted by Daniel on 2007-10-31 04:02:17