# Solution

(related to Problem: Card Triangles)

The following arrangements of the cards show

1. the smallest possible sum, $17;$ and
2. the largest possible, $23.$ It will be seen that the two cards in the middle of any side may always be interchanged without affecting the conditions. Thus there are eight ways of presenting every fundamental arrangement. The number of fundamentals is eighteen, as follows: two summing to $17,$ four summing to $19,$ six summing to $20,$ four summing to $21,$ and two summing to $23.$ These eighteen fundamentals, multiplied by eight (for the reason stated above), give $144$ as the total number of different ways of placing the cards.

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### References

#### Project Gutenberg

1. Dudeney, H. E.: "Amusements in Mathematics", The Authors' Club, 1917

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