Can you place 10 coins in 3 cups so that each cup contains an odd number of coins?
This puzzle is sometimes asked as an interview question. It also makes for a fun problem at parties and pubs.
Give it a try and watch the video for a solution.
10 coins and 3 cups riddle
Or scroll down for a text/image solution.
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Answer To 10 Coins In 3 Cups
There are only a limited number of ways that 3 positive numbers can add up to 10.
1 + 1 + 8
1 + 2 + 7
1 + 3 + 6
1 + 4 + 5
2 + 2 + 6
2 + 3 + 5
2 + 4 + 4
3 + 3 + 4
Notice that in each case at least one of the numbers is even. So it is not possible to place odd coins in each of the 3 cups.
Or think about it another way. If the cups have odd numbers 2x + 1, 2y + 1, and 2z + 1, then for the total to be 10, we must have:
(2x + 1) + (2y + 1) + (2z + 1) = 10
Which means:
2(x + y + z) = 7
We cannot have 2 times a positive number equal 7, so this is impossible.
So we need to be creative. Consider the partition 2 + 3 + 5.
Then place the middle cup with 3 INSIDE the cup with 2 coins.
Now each cup contains an odd number of coins!
In fact this solution method works for other partitions that have 1 even number and 2 odd numbers. Place the coins according to the partition, and then place one of the odd cups inside the even cup.
So we get solutions for the cases:
1 + 1 + 8
1 + 2 + 7
1 + 3 + 6
1 + 4 + 5
2 + 3 + 5
3 + 3 + 4
We cannot do this if the partition has only even numbers:
2 + 2 + 6 (can’t be done)
2 + 4 + 4 (can’t be done)
We can also place 0 coins in a cup to get a few more solutions.
0, 1, 9
0, 3, 7
0, 5, 5
Place one of the cups with an odd number of coins into the 0 cup, and once again all 3 cups will have an odd number of coins.