51¶¯Âþ

THE 100 towns, Oneton, Twoton … Hundredton, have just finished a tennis tournament in which every town played every other town once and there were no draws. In my report on the tournament I listed the towns so,

Oneton, Twoton, … Hundredton.

Then I decided I would like my list to reflect the results as follows: I would call a pair of towns X and Y a bad pair if they occur next to each other in my list, with X on the left, and Y beat X. I would then try to make a list that had no bad pairs.

Question 1: Suppose I find a bad pair in my list and I swap the two towns in the pair over. Do I necessarily reduce the number of bad pairs in my list?

I now carry out the following process:

(i) I start with my initial list.

(ii) I look for a bad pair in my list and if I am able to find one I swap the two towns over.

(iii) I take the new list I have made and do (ii).

(iv) I repeat (iii) over and over, each time doing (ii) on the list I created by the previous doing of (ii).

Question 2: Is it certain that if I carry on the process long enough I will reach a list which contains no bad pairs?

A £10 book token will be awarded to the sender of the firt correct answer opened on Thursday 30 November. The editor’s decision is final. Please send entries to Enigma 846, 51¶¯Âþ, King’s Reach Tower, Stamford Street, London SE99 0BB. The winner of Enigma 840 was Peter Mabey of Harlow, Essex.

Answer to Enigma 840

Four graduates

7558 miles