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.
At the outset, we observe that each of ABC and DEF are triangular numbers divisible by 3. Since their absolute difference consists of consecutive ordered digits, we must have:
|T(x) - T(y)| = 100*a + 10*(a+1) + (a+2), or 100*a + 10*(a-1) + (a-2), where T(x) and T(y) are triangular numbers.
or, |T(x) - T(y)| = 111*a +/- 12
or, |T(x) - T(y)|(mod 111) = +/-12 ......(i)
Again, for a triangular number T(n), the given conditions stipulate that:
100 <= T(n) <= 999
or, 200 <= n(n+1) <= 1998
or, 14 <= n <= 44 .........(ii)
But 3 divides T(n) = n(n+1)/2, and so:
n must possess the form 3t, or 3t-1 .......(iii)
We now construct a table in conformity with (i), (ii) and (iii) as
follows:
n T(n) residue of T(n)
in mod 111
14 105 105
15 120 9
17 153 42
18 171 60
20 210 99
21 231 9
23 276 54
24 300 78
26 351 18
27 378 45
29 435 102
30 465 21
32 528 84
33 561 6
35 630 75
36 666 0
38 741 75
39 780 3
41 861 84
42 903 15
44 990 102
From the above table, we observe that the valid pairs of triangular numbers (T(m),T(n)) differing by 12 in their absolute values of residues (mod 111) are:
(210, 666), (231, 465), (120, 465), (153, 276), (351, 561),
(435, 780), (780, 903), (780, 990).
Of these, the respective magnitudes of the absolute differences between the elements of each of the of the pairs (351, 561), (780, 990) do not consist of consecutive ordered digits.
Also, each of the pairs (210, 666),(780, 903) contain duplicate digits.
We denote d(x) as the digital root of x, and observe that by conditions of the problem d(ABC) > d(DEF). Thus, for the remaining pairs:
d(231) = 6 = d(465), a contradiction
d(120) = 3 < 6 = d(465), and so: ABC = 465, DEF = 120
d(153) = 9 < 6 = d(276), and so: ABC = 153, DEF = 276
d(435) = 3 < 6 = d(780), and so: ABC = 780, DEF = 435
Consequently, the required 6 digit numbers are 153276, 465120 and 780435.
Edited on August 18, 2008, 1:34 pm
Analytical Solution
