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The Four Fours Problem

Asked by Hazen Simson, Grade 12, RCS-Netherwood on October 26, 1996:
I would like to know if you know the answer to the four number problem that mathematicians have done in previous years and all mathematicians know of. It is something I have been trying for several months and without being able to get above 10 I am now searching for an answer.
I'm not 100% sure what you're asking about, but I assume (from your remarks about not being able to get above 10) that you're referring to the problem about representing integers using four 4's and operations such as addition, multiplication, etc. (If I'm wrong and you're referring to something else, such as the four-colour theorem, please let us know).

It's not really a problem "that mathematicians have done in previous years and all mathematicians know of" because it isn't really a mathematical problem. It depends on what notational convention one uses, rather than any kind of mathematical truths.

In its most basic form, the puzzle is to combine four copies of the number 4, through the basic operations of negation, addition, subtraction, multiplication, division, and exponentiation, to come up with different integers.

In this form, you can represent all the integers from 1 to 9, and some others, but you cannot represent 10. (You can prove that you can't by having a computer generate all the possible combinations and list the results).

However, there are several different ways to change the rules of the puzzle.

One way is to allow the square root symbol, so that you can take square roots without using up any additional fours (instead of having to raise something to the power of 4/(4+4) which is how you'd have to do it under the original rules).

With this, you can represent 10 (as 4 + 4 + 4/sqrt(4)). However, I don't believe you can represent 11 in this manner. (Mind you, you can't prove that 11 is impossible in the same way you can prove 10 is impossible under the original rules, because there is no limit on the number of times the square root symbol can appear in your formula, so there are an infinite number of possible expressions instead of a finite number and you can't just have a computer check them all. A proof would have to involve more advanced ideas from abstract algebra).

Another way to change the rules is to allow, not just combinations of the number 4, but combinations of numbers that can be written in decimal notation using the digit 4 (for example, 44 or .4). With these rules, you can represent 10 = 4/.4 + 4 - 4, 11 = 44/4 + 4 - 4 (or 11 = 4/.4 + 4/4), and 12 = (44 + 4)/4. However, you cannot represent 13.

Another way to change the rules is to allow all of the above. This is probably the form of the puzzle you are working on. You still can't generate all integers, but you can go quite a bit past 13. The previous examples should help you figure out how.

Finally, you could change the rules to allow functions such as logarithms to occur in your expression. If you do that, then every integer can be expressed. For example:

        1 = - log           log   sqrt(4)
                 4/sqrt(4)     4
        2 = - log           log   sqrt(sqrt(4))
                 4/sqrt(4)     4
        3 = - log           log   sqrt(sqrt(sqrt(4)))
                 4/sqrt(4)     4

and so on. (See if you can prove this!)

In summary, this puzzle depends entirely on what rules you choose. It has more to do with manipulating the symbolism in clever ways than on any mathematical truths, and if mathematical notation had evolved differently, the outcome of the puzzle would be quite different too.

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