Envelope Packing Puzzle

Tee Jet Whaw (Imperial Maths Competition 2020 Team Round)

There are a total of 6 envelopes of different sizes. An envelope can contain one or more smaller envelopes. How many ways are there to pack the 6 envelopes into a parcel?

Scroll down for solution!

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Solution

Don’t break into cases! Let An be the n-th largest envelope.

Starting from A1, there’s one choice to arrange it. Next, take A2. It can be squeezed into A1 or placed next to it, giving a total of 2 choices. Now, we have 3 choices for A3: directly contained by A1, directly contained by A2, or standalone. This is true regardless of the choice made with the placement of A2. Following similar logic, A4 has 4 options.

As we proceed, things start to get a little complicated… For example, how do we persuade ourselves that there are 6 options of where to place A6, when A1– A5 are already in place? Consider the following.

  • There are at least 6 options:
    • A6 can only be directly contained in A1, A2, A3, A4, A5, or nothing. These are all different cases.
  • There are at most 6 options:
    • If there is the 7th option, then what envelope is outside of A6? It must be one of A1-A5 or nothing at all. We’ve already considered these cases above, so in fact there cannot be any new cases.

Hence, in total we have 6! = 720 options.

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