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
(In reply to
that was easy by alison)
Alison, you are doing exactly what the problem states, but the problem asks you to prove why the total is 111. The example you showed is just that, one of the many examples you can find.
For example, doing this with the opposite diagonal you chose gives 1+8+15+22+29+36=111
Now, something I noted from the two examples is this. The difference between the numbers cancels each other out.
6 11 16 21 26 31
1 8 15 22 29 36
The differences are 5, 3, 1, -1, -3, -5.
Also, the sequence Alison proposed is x, x+5, x+10, x+15..., while the one in the opposite diagonal is x, x+7, x+14, x+21...
So x+x+5+x+10+x+15+x+20+x+25=111, then x=6. Similarily, when x+x+7+x+14+x+21+x+28+x+35=111, math says x=1.
If you were to pick all from the same line as the problem states that you can't, then x+x+6+x+12+x+18+x+24+x+30=111 would mean that x=3.5, which isn't an integrer.
Therefore for every number there's a total value which can be added to 6 times itself to equal 111.
For 1, it's 105. 105 can be found by the sum of 7n from n=1 to n=5.
For 6, it's 75. 75 can be found by the sum of 5n from n=1 to n=5.
Now let's take a look at 2. In order to to this with two you have to add 1 to the diagonal you used in 1, but you'd have to substract 5 from the last value (36-5=31), because that's the only number left.
In other words, as you keep crossing off columnd and rows, all that's left is diagonals. The numbers in the diagonals differ by 5 or 7 depending on which direction you take. However, since columns and rows are crossed out, there is no way to select numbers from the same row or column, and you're forced to either add or substract 7 to TWO numbers and add and substract 5 to TWO numbers, therefore adding NOTHING to the total or 111 you can find in the diagonals.
OK, my explanation is too complicated. There has to be a simpler way.
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Posted by Alexis
on 2005-11-08 02:43:02 |
re: that was easy (and thoughts)
