Arrange two of each digit (0-9) into a 20 digit long number, which does not begin with a zero, then score your number as follows:

For every 2 digit sequence that forms a perfect square score 2 points, for every 3 digit sequence that forms a perfect square score 3 points. A four digit square would score 4 points, and so on.

For example, if your number was 58738219024971503664, then you would score 2 points each for 49, 36, and 64 and 6 points for 219024 giving a total of 12 points. You may not count 036 as a three digit square.

What is the maximum number of points you can score?

I solved this problem for bases 2 and 3 by brute force (by hand)

In base 2 there are only 4 digits and only two perfect squares (100 and 1001)

Of the six possible numbers, the best is 1001 which is worth 7 points.

 

In base 3 there are 6 digits and twelve perfect square with only 2 of each digit.

Of the 90 possible numbers, the best is 221001 which is worth 17 points.
The perfect squares are 100, 221, 22100, 221001

 

Does this help us with the base 10 problem?   I'm not sure, but it does hint that the 0's should go together, but need not be at the end.

Posted by Jer on 2005-03-30 20:10:05