Sunday Times Teaser Teaser 2637 – Powerful Team

by Danny Roth

The players in a local football team have the numbers 1 to 22 with eleven playing and eleven in reserve in any match.

In one match, the players on the field were such that there was no sequence of three consecutive numbers among them.  The same was true of the reserves.  The sum of the fielded player’s numbers was also a perfect square.

All these facts remained true after a single substitution at half time but the perfect square was then higher.

What were the numbers (in increasing order) of the players in the first half of the match?

Here is my solution:

Here is my analytical solution:

1,2,4,5,7,8,10,11,13,14,16=91 giving us the minimum value of the first half team as 100 and taking into account of the fact that the maximum diffrerence in the sum of the teams for both halfs cannot be larger than 21(22-1) which gives us the sum of the second half half team as 121.The minimum sum of the first half team numbers can be found by trial and error.: