# SEVEN-COUNTERS GAME

This simple puzzle is from the 1908 book “A Scrap-Book of Elementary Mathematics” by William F. White.

Required to place seven counters on seven of the eight spots in conformity to the following rule: To place a counter, one must set out from a spot that is unoccupied and move along a straight line to the spot where the counter is to be placed.

Clarification (from GentlePurpleRain):

For each counter, you must pick an empty spot, then move along a line to another empty spot where you place the counter.

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• The problem description is not well-worded. I first interpreted it to mean that you pick a spot that is unoccupied (any of them), and move along the lines, placing counters. That makes it an extremely trivial problem. But I think the intention is that for each counter, you must pick an empty spot, then move along a line to another empty spot where you place the counter. That makes the problem slightly less trivial (although a solution becomes apparent to me fairly quickly). Oct 30, 2023 at 14:51
• @GentlePurpleRain Thanks for your comment. I have added a clarification to the wording of the puzzle. Oct 30, 2023 at 19:24
• @GentlePurpleRain to be honest perhaps your two interpretations are equivalent because the solutions are [redacted to avoid spoilers] Oct 30, 2023 at 21:09

## 1 Answer

In my representation, unoccupied points are represented by letters and occupied points are represented by counters (numbers):

  A  B
H      C
G      D
F  E


Step 1:

Start with any unoccupied point: say A. A links to both D and F. Set out from a spot that is unoccupied (A) and move along a straight line to the spot where the counter is to be placed (F):


A  B
H      C
G      D
1  E


Step 2:

F is linked both from A and C. Since you've already came from A, the strategy is to set the next counter in C. Set out from a spot that is unoccupied (H) and move along a straight line to the spot where the counter is to be placed (C):


A  B
H      2
G      D
1  E


Step 3:

C is linked both by F and H. Since F is already occupied by 1, the strategy is to set the next counter in H. Set out from a spot that is unoccupied (E) and move along a straight line to the spot where the counter is to be placed (H):


A  B
3      2
G      D
1  E


Step 4:

H is linked both by C and E. Since C is already occupied by 2, the strategy is to set the next counter in E. Set out from a spot that is unoccupied (B) and move along a straight line to the spot where the counter is to be placed (E):


A  B
3      2
G      D
1  4


Therefore,

following this pattern, you'll get as final result:


A  5
3      2
6      7
1  4