UKMT Pink Kangaroo, 2016

Combinatorics Difficulty 2.0 Junior Find the answer United Kingdom

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) $6$

We don't reproduce this publisher's solutions. Their own solution is here — work the problem first.

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: UK Mathematics Trust, licensed © UK Mathematics Trust; past papers published free at ukmt.org.uk. Statement reproduced verbatim; metadata (topic, difficulty) added by this project. Solutions are the publisher's, linked not copied.