There are three 6 digit numbers with the following properties applicable to each:

1. All digits are unique.
2. The first three digits ABC form a triangular number as do the latter three, DEF; both are multiples of 3.
3. The digital root/sum of the first triangular number is greater than that of the second.
4. Three consecutive digits form the difference of the triangular numbers, either being ascending or descending.

Identify the three 6 digit numbers.

There are thirty-one 3-digit triangular numbers:

105, 120, 136, 153, 171, 190, 210, 231, 253, 276,
300, 325, 351, 378, 406, 435, 465, 496, 528, 561,
595, 630, 666, 703, 741, 780, 820, 861, 903, 946,
990


Of these thirty-one, the following ten are not divisible by 3:

136, 190, 253, 325, 406, 496, 595, 703, 820, 946


Eliminating the above ten, of the remaining twenty-one, four do not have unique digits:

171, 300, 666, 990


Eliminating the above four, of the seventeen 3-digit numbers remaining, seven pair have the property that their difference is a 3-digit number where its digits are in consecutive sequence:

561 - 351 = 210
561 - 105 = 456
465 - 120 = 345
276 - 153 = 123
465 - 231 = 234
780 - 435 = 345
903 - 780 = 123


As 561 shares both the digits 1 and 5 with 105 and 351; and 903 shares the digit 0 with 780; these three pairs are removed from consideration. The digital root sum of each number of the remaining four pairs are as follows:
S[465] = 6   S[120] = 3
S[276] = 6   S[153] = 9
S[465] = 6   S[231] = 6
S[780] = 6   S[435] = 3


As 465 and 231 share the same digital root sum, this pair is removed from consideration, leaving the following three pairs (each number of each pair being listed in descending sequence of digital root sum value):
465, 120
153, 276
780, 435


Thus, the three 6-digit numbers are as follows:
465120, 153276, and 780435

Edited on August 21, 2008, 8:34 pm

Posted by Dej Mar on 2008-08-18 01:50:07