3 cards

Probability Level 1

Alan wrote 6 different numbers, one on each side of 3 cards, and laid the cards on a table. The sums of the two numbers on each of the three cards are equal. The three numbers on the hidden sides are prime numbers. If the visible numbers are 44, 59 and 38, what is the average of the hidden prime numbers?

14 17 15 16

Elliott Chen by Elliott Chen , 14, Canada

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2 solutions

Tin Le
May 19, 2019

We know that 44 and 38 are even numbers and 59 is an odd number.

Since the sum of the numbers on each of the 3 cards are equal, we can deduce that there are 2 even numbers and 1 odd number, or 1 even number and 2 odd numbers on the hidden sides of the card.

The only prime number which is even is 2. Therefore, there must be an odd number on each hidden sides of the number 44 and 38 and the number 2 is on the hidden side of the number 59.

59 + 2 = 61. Hence, the number on the hidden side of the number 44 is 61 - 44 = 17; the number on the hidden side of the number 38 is 61 - 38 = 23.

The average of the prime numbers is 2 + 17 + 23 3 = 14 \frac{2+17+23}{3}=\boxed{14}

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Eric Scholz
May 19, 2019

Since two numbers are even and one is odd, there has to be even and odd prime numbers on the other side -> 2 pairs with the odd number, making the sum 59+2=61 61 = 44+17
61 = 38+23

23+17+2=42
42/3=14

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