Notice that in this arrangement 1 2 3
4 5 6
7 8 9
opposite values add up to 10: 1 + 9, 2 + 8, 3 + 7, 4 + 6.Why these numbers do this is simply because the sequence is laid down in left to right, top-to-bottom order in a symmetric way, and these values pairs end up equidistant elements from the opposite ends of the sequence. If you put your index fingers on 1 and 9, and then count upwards with the left one and down with the right, you can see how this plays out. And this summing to 10 to do with why the last digit of the sums you are seeing is always 0: 3330. For instance in 123 + 321 + 456 + 654 + 789 + 987, the last digit is (3 + 1) + (6 + 4) + (9 + 7). We can rearrange these six numbers into (1 + 9) + (4 + 6) + (7 + 3) = 10 + 10 + 30 = 30. Ok, so now we have a 0, and a carry of 3. Next, note that since the diagonally opposite elements add to 10, all the three-element traces that pass through the center 5 necessarily add up to 15: (1 + 5 + 9) = (2 + 5 + 8) = (3 + 5 + 7) = (4 + 5 + 6) = 15. In calculating the second digit of the sum you have 2 5 and 8, which occur twice: (2 + 2 + 5 + 5 + 8 + 8) = 30. Combine that with the carried 3 and you get 33. Put down the 3 and carry the 3. Then again, the 100's digit is just mirror image of the ones: it adds up to 30, which combines with the carried 3 to make 33. With 784 + 487 + 159 + 951 + 623 + 326, though you have rearranged the digits to form corner triangles, that is just a red herring. If you look at the ones digits inside this sum, you have 4 7 9 1 3 6. These is just the set made up of the left and right columns of the square, which we know can be put into 3 pairs adding to 10, making 30. Again we get our 0 to put down and 3 to carry. The middle digits, the tens, are once again 8 5 and 2, doubled up again: another 15 x 2 = 30: put down 3, carry 3. And so it goes. |