February 2024
Problem of the Month

Chess Tournament
Submitted by Mümtaz Keskin
from Turkey



Keloğlan visits a farm where 16 farmers live and organizes a chess tournament for all the farmers.

In the tournament, each pair of farmers plays exactly one match against each other.

Keloğlan gives 5 coconuts to each farmer for every match they win and 2 coconuts for every match that ends in a draw.

Farmers do not receive any coconuts for matches they lose.

After the tournament, Keloğlan realizes that he has distributed a total of exactly 550 coconuts to the farmers.

Based on this, what is the maximum number of farmers who did not have a draw in the tournament?

Note: Keloğlan is a well-known character in Turkish folklore.   He is a clever and lucky character.   He represents the Anatolian people who can have big dreams, who are virtuous, prudent, a little bald, a little romantic and very sporty.


Solution:

The answer is 5 farmers.

Since there are 16 farmers and each must play the other, that results in
16C2 = 120 chess matches.

If there were 120 winners and no draws, then that results in 600 coconuts, but Keloğlan only distributed 550 coconuts.

So, trying several combinations, the one that works is 70 wins and 50 draws.

The 70 wins yields 350 coconuts and the 50 draws result in 200 coconuts (because 2 farmers get 2 coconuts for each draw).

If ten farmers draw with each other (leaving six other farmers), that results in only 45 draws.   So, one other farmer must have some draws.   Therefore the maximum number of farmers that did not have a draw is five.



Correctly solved by:

1. K. Sengupta Calcutta, India
2. Ivy Joseph Pune, Maharashtra, India
3. Kamal Lohia Holy Angel School,
Hisar, Haryana, India
4. Kelly Stubblefield Mobile, Alabama, USA
5. Colin (Yowie) Bowey Beechworth, Victoria, Australia



Send any comments or questions to: David Pleacher