You may split the numbers however you like. Once the two groups are sorted in opposite directions, the final sum no longer depends on your choice.
Two Columns
Riddle statement
Write down the numbers from 1 to 10 and divide them however you like into two groups of five.
Then:
- sort one group from smallest to largest;
- sort the other from largest to smallest;
- place them in two columns, forming five rows;
- in each row, subtract the smaller number from the larger;
- add the five differences.
What will the final result always be?
Show solution
Solution
Call 1, 2, 3, 4, and 5 the small numbers, and 6, 7, 8, 9, and 10 the large numbers.
Number the rows from 1 to 5, from top to bottom. Suppose row i contained two small numbers. In the ascending column, the first i entries would all be small. In the descending column, the last 6 − i entries would also all be small. That would require at least i + (6 − i) = 6 distinct small numbers, but only five exist.
The same argument shows that no row can contain two large numbers. Each row therefore contains exactly one small number and one large number.
When the five differences are added, every large number appears with a positive sign and every small number with a negative sign:
(6 + 7 + 8 + 9 + 10) − (1 + 2 + 3 + 4 + 5)
40 − 15 = 25.
The initial division changes the pairings, but not the total.