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So my friend made a square like this with this amount of cards in each pile

4 3 2 1
3     2
2     3
1 2 3 4

He then said that each line, horizontal and vertical, adds up to 10 cards. Then he asked me to pick a card from the cards left over after making the square and asked me to put it on any of the piles in the square. He then rearranged the cards and all the lines still added to 10 cards on each pile, how was it possible?

Henry
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Boo
  • 21

1 Answers1

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You have $30$ cards in the piles rather than $4\times 10 = 40$ because the corner piles $4+1+1+4$ are each on two sides and are counted twice

So when you pick an extra card, one of the piles gets one bigger and then you have $31$ cards in the piles and you need to re-arrange the piles so the corner piles add up to $9$ while each side still adds up to $10$

As an example, if you change one of the piles of $1$ to become a pile of $2$ then first of the following arrangements of the new piles works, and similarly for the other possible changes

1 -> 2    2 -> 3    3 -> 4     4 -> 5

4 3 2 1   4 3 2 1   4 3 2 1    4 3 2 1  
2     3   3     2   3     2    3     2
2     4   2     4   2     4    1     5
2 3 3 2   1 3 3 3   1 2 4 3    2 3 3 2
Henry
  • 157,058