A 3 by 3 grid contains 9 duodecimal digits, thus creating a crossword-like puzzle.
The numbers are defined as follows:

ACROSS
1. In its decimal form a square of this number is a concatenation of 2 consecutive integers.
2. A "taxicab number" minus 2
3. This prime number remains prime (in its decimal form) if augmented by its reversed form.

DOWN
1. The duodecimal year A.D. in which St Columba allegedly met the Loch Ness monster.
2. This number remains prime read both from left to right and right to left (in its decimal form).
3. Most of the year 2008 A.D. (Gregorian calendar) corresponds to this year number in Islamic calendar.

Task1: Restore the contents of the grid.
Task2: Write the value of the diagonal (reading from the top right to the bottom left) in the bases 2,3,4 and 7 and comment upon the results.

The answers to 1 down and 3 down are straightforward (though I had to google them both).  These sufficiently restricted the possibilities for the remaining answers and I obtained the following grid:

3 b 9

b b b

1 7 1

The diagonal from top right to bottom left is 9b1, or 1429 in base 10.  In the other bases this is:

Base 2: 10110010101

Base 3: 1221221

Base 4: 112111

Base 7: 4111

At first glance (assuming these are right), I'm not sure what these have in common to comment upon.  If you consider these representations in base 10, they're all prime except for 10110010101.  I'll either have to check my work to see if I made a mistake somewhere, or think harder about what kind of commentary the problem is looking for. 

 

Posted by tomarken on 2014-04-02 13:25:52