PK 2016 Q14
14. Eight unmarked envelopes contain the numbers: 1, 2, 4, 8, 16, 32, 64, 128. Eve chooses a few envelopes randomly. Alie takes the rest. Both sum up their numbers. Eve's sum is 31 more than Alie's. How many envelopes did Eve take?
- A 2
- B 3
- C 4
- D 5
- E 6
Pink Kangaroo 2016, Question 14. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D 5
The sum of the numbers in the envelopes is 255. Let \(E\) be the sum of Evie's numbers. Then Alie's numbers will have a total of \(E - 31\). Hence we have \(E + (E - 31) = 255\), giving \(2E - 31 = 255\), so \(2E = 286\) and \(E = 143\). The only way to add up powers of two to get 143 is 128 + 8 + 4 + 2 + 1, so Evie took 5 envelopes.
Solution © UK Mathematics Trust, from the PK 2016 solutions.