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Six Digits = Triangle + Triangle (Posted on 2008-08-17) Difficulty: 3 of 5
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.

See The Solution Submitted by brianjn    
Rating: 3.0000 (1 votes)

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Solution Analytical Solution Comment 3 of 3 |

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
  Posted by K Sengupta on 2008-08-18 12:21:21

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