Each different capital letter below represents a different digit. Each occurrence of the same capital letter represents the same digit.

ZWEI is a prime number.
DREI is a triangular number (that is, of the form n*(n+1)/2)
VIER is a perfect square.

What do DREI and VIER represent?

Added logic bonus:

If I told you there were no zeros anywhere, and I also told you that giving you the value of Z would allow you to figure out what ZWEI represented, what would you say the value of ZWEI is?

At first glance, it is clear that I is an odd number since ZWEI is prime.  Since I is odd, DREI must also be odd.  This means that in the formula for DREI, n*(n+1) cannot be divisible by 4.  Thus, n follows the series {1,2,5,6,9,10,13,14, . . .}.  Also, the value of n falls in the range { 44 < n < 139}.

Using a spreadsheet, all the triangular numbers for these values of n can be compiled.  One will find that each number ends in either 1, 3, or 5.  By eliminating those numbers with repeating digits, the list of possible values for DREI is reduced from 48 down to 24 numbers.

Next, find the possible values for VIER.  This is done by computing the squares of every integer between 32 and 99.  This list is then shortened by removing those with repeating digits as well as those containing values other than 1, 3, or 5 in the hundreds digit.  The list of possible values for VIER is thus reduced from 68 numbers down to 7.

A comparison of these lists finds only one match for REI-IER.

DREI = 1653

VIER = 4356

We now have half of ZWEI solved. (ZW53)

The values for Z & W must come from the set {0,2,7,8}.  Even accounting for zero, there are 12 possible combinations of digits to form ZWEI.  From this set, the following are prime:

{0853,2053,2753,7253,7853,8053,8753}

Eliminating those values containing zeros yields:

{2753,7253,7853,8753}

If it can be determined which value is true by knowing Z, then the value of ZWEI is either 2753 or 8753.

Posted by hoodat on 2007-06-22 14:33:39