GMAT Math : The box principle

The box principle is a powerful arithmetic tool and you never know when it may come in handy in the GMAT Quant section.

Let’s start with an example :

Bob works in the local post department is placing a set of cards kept in a sack in envelopes. His KPI is such that each card has to eventually go in an envelope and he can choose to put multiple cards in the same envelope. Given there are 137 cards, what is the maximum number of envelopes that should be with him to ensure there is at least 1 envelope with a minimum of 3 cards?

A) 68

B) 72

C) 59

D) 52

E) 69

The pigeonhole principle states that if there are k boxes and m items such that :

m = nk + 1 where m, n and k are positive integers, then there is at least 1 box with n + 1 items

Lets use a simple example – imagine there are 3 boxes and 4 socks – no matter how you distribute the socks across the 3 boxes, there will be a box with a minimum of 2 socks!

If we were to represent it as the pigeonhole equation, then 4 (items) = k (3 boxes) + 1.

k = 1 and hence there is 1 box with k + 1 (i.e., 2) items

Coming to our question,

We know k + 1 = 3 (minimum 3 cards in an envelope), hence, k = 2

Number of items = m = 137

Plugging into our equation, m = nk + 1 and hence 137 = 2n + 1

n = 68

The answer is (A). Cool stuff – isn’t it 🙂