🦘 Math Kangaroo Grade All Felix 1-2 Ecolier 3-4 Benjamin 5-6 Kadett 7-8 Junior 9-10 Student 11-12 ⇄ switch contest
2014 Math Kangaroo

Problem 16

Problem 16 · 2014 Math Kangaroo Medium
Number Theory dice-facesprimessum-constraint

If you add the numbers on opposite faces of this special die, you get the same total three times. The numbers on the hidden faces are prime numbers. Which number is on the face opposite to 14?

Figure for Math Kangaroo 2014 Problem 16
Show answer
Answer: E — 23
Show hints
Hint 1 of 2
Opposite faces all add to the same total; the three hidden faces are primes.
Still stuck? Show hint 2 →
Hint 2 of 2
Use the three visible numbers 18, 35, 14 and make each opposite a prime with one common sum.
Show solution
Approach: find the common opposite-sum that makes all hidden faces prime
  1. Let every opposite pair add to S. Then the hidden faces are S−18, S−35 and S−14.
  2. S = 37 works: hidden faces 19, 2 and 23 are all prime.
  3. The face opposite 14 is 37 − 14 = 23.
Mark: · log in to save
↑