Four Fours Problem

Here’s a challenge that you may wish to try: can you express all the numbers from 1 to 100 using an combination of only four 4’s?

The operations and symbols that are allowed are: the four operations (+,x,-,/), concatenation (44 is ok and uses up two 4’s), decimal points (using 4.4 is ok), powers (using 44 is ok), square roots, factorials (using 4! is ok), and overbars for indicating repeating digits (e.g., writing .4 with an overbar would be a way of expressing 4/9). Ordinary use of parentheses are allowed. No digits than 4 are allowed.

So for instance, here is a complicated way to write 70 using four fours:4 + [44/Sqrt(.4-overbar)].

This problem is sometimes called the four fours problem. I’m not sure where it first originated but it was popularized by Martin Gardner, among others.

Presentation Suggestions:
This makes an excellent extra credit problem. Or, you might suggest it as a joint project for a whole class to work on: have them post solutions on a bulletin board as they find them. For a course it makes a nice programming exercise in the language prolog.

The Math Behind the Fact:
Actually, all the numbers less than 113 can be constructed in this fashion. While I won’t spoil the fun and tell you the answers, let me just say (from experience) that the hardest numbers to express in four 4’s are the numbers 69 and 73. These require especially clever combinations of the operations above.

A difficult (and as far as I know unsolved) mathematical challenge is to prove that the number 113 cannot be constructed using these operations.

It should be noted that there are many versions of this problem that have floated around, differing only in the sets of operations that are allowed. (For instance, 113 can be done if you allow arccos as a function.)

How to Cite this Page: 
Su, Francis E., et al. “Four Fours Problem.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

Fun Fact suggested by:
Francis Su

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