Each of the capital letters in bold in this 3x3 grid is to be substituted by a different digit from 1 to 9 such that :
- Each of A, C, G and I is even, and:
- The sum of the digits in each of the four 2x2 subgrids is equal.
A B C
D E F
G H I
Disregarding the rotations and reflections, how many distinct solutions are there?
Since the sums of the subgrid elements are equal, and they share common elements:
the lower subgrids give D + G = F + I (1)
the upper subgrids give A + D = C + F (2)
the left subgrids give A + B = G + H (3)
Equations (1) and (2) give A + I = C + G, and since these are the even digits, 2, 4, 6, 8, it follows that 2, 8 and 4, 6 must be at opposite corners.
Without loss of generality, but to avoid including reflections and rotations, take A = 2, C = 4, G = 6 and I = 8.
(1) now gives D = F + 2 with the possible results: D F
3 1 (1.1)
5 3 (1.2)
7 5 (1.3)
9 7 (1.4)
(3) now gives B = H + 4 with the possible results: B H
5 1 (3.1)
7 3 (3.2)
9 5 (3.3)
Result (3.2) is incompatible with all of (1.1) to (1.4) since 3 or 7 would then be duplicated.
Similarly, (3.1) is compatible only with (1.4) giving 2 5 4
9 3 7
6 1 8
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and (3.3) is compatible only with (1.1), giving 2 9 4
3 7 1
6 5 8
So, there are two distinct solutions (agreeing with previous postings).
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Posted by Harry
on 2009-08-18 22:32:50 |
Analytical Solution
