3   10    7
4     9    5
6     5    1
7     2    5
8     1    7

Total:   25

I have no imagination at the moment...

1  10  9
2  9    7
3  8    5
4  7    3
5  6    1
         --
         25

I'm annoyed at this because at a glance it seems easy to prove how it works, but on closer inspection it's not at all!
That or I'm being dumb...

Zach wrote:

I'm not sure if I got that correct, based on the point of the idea. Perhaps I got it wrong.

The first column should be in ascending order.

mathsyperson wrote:

I'm annoyed at this because at a glance it seems easy to prove how it works, but on closer inspection it's not at all!

I agree; it looks trivial, but it's not!

Maths: It is still amazing to me how something that starts so simply (one pebble plus another pebble makes two pebbles) can become so rich and varied so quickly.

I mean, you can make a circle by standing on one foot and dragging the other one around on the sand. Who could possibly think that comparing the distance your feet are apart to the length your foot travels would be a number that has digits going on forever ...

Simply inspires awe.

I got into Excel and ran the first one for 50 terms (up to-4/99 ) and it got to 3.1216 (0.02 off)

i am well confused...

They were exchanging information about pi 'doccuments'..........

Here's a hint as to why the total is always 25.

In the examples above, color or shade (in the first two columns) the numbers 1, 2, 3, 4, and 5.  Do you see a pattern?

1 10 9
2 9 7
3 8 5
4 7 3
5 6 1

25

1 20  19
2 19  17
3 18  15
4 17  13
5 16  11
6 15   9
7 14   7
8 13   5
9 12   3
10 11 1

That's 100!

Re-arranging,
1 17  16
2 20  18
3 13  10
4 19  15
5 16  11
6 18  12
7 15   8
8 14   6
9 11   2
10 12  2
100 yet again!

Should it be always (n/2)² for 0 to n???
(Just like the side total of a magic square containing numbers 1 to n² is
(n³+n)/2 ??? )

Last edited by ganesh (2005-07-19 15:03:46)

Ganesh has extended the idea.

It seems logical. Let's try a 6 row one (1 to 12):

1   12   11
3   10    7
4     8    4
5     7    2
9     6    3
11   2    9

Total = 36

That is 6²

1 4 3
2 3 1
Total = 2²

1 4 3
3 2 1
Total = 2²

1 3 2
4 2 2
Total = 2²

2 4 2
3 1 2
Total = 2²

2 3 1
4 1 3
Total = 2²

3 2 1
4 1 3
Total = 2²

Is that all of them?