Someone fills a 6x6 matrix with the numbers from 1 to 36 first across, then down, like:
1 2 3 4 5 6
7 8 9 10 11 12
13 14 15 16 17 18
19 20 21 22 23 24
25 26 27 28 29 30
31 32 33 34 35 36
Now they ask for volunteers to randomly select numbers. The number selected will be circled and the others in the same row and column will be crossed out. Non-crossed out numbers are selected until no more numbers can be chosen. For example, selecting 8 means that 7, 9, 10, 11, 12, 2, 14, 20, 26, and 32, can no longer be chosen.
When all is said and done, the total of the circled numbers is 111.
Can you prove why this is so?
To give credit where due, I first saw this on curiousmath.com
An element of the array is A(I,J) = 6(I-1)+J.
There is only one circled element in each row
and column. Summing the element in each row:
Row Element
1 0 + J_1 +
2 6 + J_2 +
3 12 + J_3 +
4 18 + J_4 +
5 24 + J_5 +
6 30 + J_6
= 90 + (J_1 + J_2 + J_3 + J_4 + J_5 + J_6)
= 90 + 21
= 111
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Posted by Bractals
on 2005-09-22 02:57:49 |
Solution
