December 2019
Problem of the Month

Kopek Problem
by Boris Kordemsky in GAMES

This puzzle uses 17 Russian coins:
Five 20 kopek coins
Three 15 kopek coins
Three 10 kopek coins
Six 5 kopek coins

Place a coin in each square of the diagram so that the number of kopeks along each straight line is 55.

Solution to the Problem:

There are many solutions.   Here are two of them:





James alarie sent in his excellent analysis of the problem:

The only way to have three coins add to 55 is by using two 20's and a 15.
You need a 15 for each of the four sides but only have three 15's, so one of them must be in a corner.
Four of the 20's are used up on the two sides adjoining that corner 15, so you only have one left for the other two sides.   It must be in the corner diagonally opposite that corner 15.
The two remaining 15's have to go in the empty slots on the sides.
The diagonal already having 20's at it's end points needs three coins to fill it's slots.   The only way to do that is with three 5's.
That leaves six empty slots for the three 10's and remaining three 5's.   There are eight ways of putting them in to add to 55 as needed, but two of those are reflections of others.
There are six unique solutions not counting rotations and reflections.

Counting rotations and reflections, there are thirty-two solutions.
One corner must be a 15 coin and it can be any one of the four rotational positions.
Once that 15-corner is set, the remainder of the outer ring is set and does not change through rotation nor reflection.
The (20,5,5,5,20) diagonal is forced.
There are eight ways of placing the remaining coins to satisfy the 55-total rule.   This eight accounts for all rotations and reflections.
Eight times four is thirty-two.

Correctly solved by:

1. Soonho You	JeonJu, South Korea
2. James Alarie	Flint, Michigan
3. Ivy Joseph	Pune, Maharashtra, India
4. Alyxa Martin	Mountain View High School, Mountain View, Wyoming
5. Kendall Carr	Mountain View High School, Mountain View, Wyoming
6. Aubree Andre	Delta High School, Delta, Colorado
7. Gauge Lockhart	Delta High School, Delta, Colorado
8. Katelynn Hope	Mountain View High School, Mountain View, Wyoming
9. Cortlin Condos	Mountain View High School, Mountain View, Wyoming
10. Skyler Kraai	Delta High School, Delta, Colorado
11. Kenzy Kellum	Mountain View High School, Mountain View, Wyoming
12. Adam Bates	Mountain View High School, Mountain View, Wyoming
13. Breckin Barnes	Mountain View High School, Mountain View, Wyoming
14. Izabell Hickey	Mountain View High School, Mountain View, Wyoming
15. Braeden Walk	Mountain View High School, Mountain View, Wyoming
16. Aubrey Reger	Mountain View High School, Mountain View, Wyoming
17. Martin Corral	Delta High School, Delta, Colorado
18. Scott Wells	Delta High School, Delta, Colorado
19. Tara L. Fiedler	Goose Creek CISD, Baytown, Texas
20. Figen Furtun	Istanbul, Turkey
21. Brijesh Dave	Mumbai City, Maharashtra, India
22. Kelly Stubblefield	Mobile, Alabama

Send any comments or questions to: David Pleacher