Each number in the list is to be converted into a base b, b = 2 through 10, with each base being used only once. Once in the appropriate base, the number will have a square number of digits, and the only characters will be 0 and b-1. The number of digits are as follows:

(base b)^(# of digits in base b) = (# of digits in base 10) 2^361 = 112 3^225 = 113 4^196 = 122 5^169 = 119 6^169 = 132 7^144 = 121 8^144 = 131 9^121 = 116 10^121 = 121

Once in the appropriate base, the number can be arranged as a square to display some figure (each figure is given below). Each figure is a number in some writing system, all of which can be found at omniglot.com. The number the corresponds to a letter by a simple A=1, B=2, etc. substitution, and the ordering of the resulting letters to form the answer is given by the bases the original numbers were converted to:

letter numeral base square size writing system A 1 2 19x19 Financial Chinese C 3 3 15x15 Javanese A 1 4 14x14 Thai D 4 5 13x13 Tibetan E 5 6 13x13 Tengwar M 13 7 12x12 Mayan I 9 8 12x12 Lao E 5 9 11x11 Arabic S 19 10 11x11 D'ni

The figures are as follows (they all look better with a 1:1 aspect ratio, like you'd get using graph paper...):

base two:

.........0......... .........0......0.. .00000000000000000. .........0......... .....000000000..... ..0................ ..0000000000000000. ..0.............0.. .0.............0... .0...000000000..... ................... .....000000000..... .....0.......0..... .....0.......0..... .....000000000..... .....0.0...0.0..... ........0.0........ ........0.0........ 0000000000000000000

base three:

..0...0.0...... .0.0.0.0.0.000. 0..0.0.0.0.0.0. 0..0.....0.0.0. 0..0.....0.0.0. 0..0..0..0.0.0. .0.00000.0.0.0. .........0.0.0. .....00000.0.0. ....0......0.0. ...0.......0.0. ...0.......0.0. ...0.......0.0. ....0.0000.0.0. .....0...00..00

base four:

.....0000...... ...0.....000... ..0.........0.. .0...........0. .0...........0. 0.............0 0....0000.....0 0....0..0.....0 .0...0..0.....0 .0...0000....0. ..0....0....... ..0....0....... ...0...0....... ....0000.......

base five:

......00..... ....00.0..... ..00...0..... 00......0.... 0........00.. 0..........0. .0........... .0........... ..0.......... ...00........ .....00...... .......00..0. .........00..

base six:

.0........... .0........... .0........... .000000000000 .0...0....... .0...0....... .0...0....... .0...0....0.. .0....0000... ..0.......... ..0.......... ...0....0.... ....0000.....

base seven:

............ .000.000.000 .000.000.000 .000.000.000 ............ ............ 000000000000 000000000000 ............ 000000000000 000000000000 ............

base eight:

...000...0.. ..0...0...0. .0....0....0 .0....0....0 0.....0....0 0.....0....0 0.....0....0 0.....0....0 0.....0....0 000...0...0. 0.0...0...0. 000....000..

base nine:

..0000000.. ..0........ ..0........ ..0........ ..00000.... .......0... ........0.. ........0.. ........0.. ..0....0... ...0000....

base ten:

........... 00000000000 .0.......0. .0...00000. .0...0...0. .0...0...0. .00..0..00. .0.0.0.0.0. .0..000..0. 00000000000 ...........

*2006 MIT Mystery Hunt*