Do you like challenges? Do you often pride yourself on your mathematical brilliance? Do you always act indignant or chagrined, but secretly love the accusations of being a nerd? If yes, then this challenge is for you.

The four fours problem was published in the “Mathematical Recreations and Essays” written by W.W Rouse Ball in 1914, who nonchalantly described the problem as “tradition recreation”. However, the whimsical puzzle is known to have been first conjured in the 19th century, and its popularity was at its zenith around 1890. Gradually, it grew less well known, until Rouse published his essays and revived interest in the puzzle.

The puzzle slowly grew into an intellectual pastime. Even Paul Dirac, the Nobel Prize-winning physicist who discovered anti-particles and is widely lauded for his mathematical genius alongside Richard Feynman, gave it a try and, as one would expect, he solved it. Or, rather, as we’ll eventually find out, he *hacked* it.

## The Rules

The rules are simple: create every whole number from 0 to possibly infinity using mathematical expressions that only consist of four fours manipulated by any number of mathematical operations. For instance, can be expressed as , while can be expressed as

However, the four basic operations can only create a handful of numbers. Therefore, we must turn to other operations. This would include concatenation or the combination of two numbers to form a larger number such as*to infinity!*

Some even resort to summoning absurd or atypical operations, such as the percentage or even permutation and combination (4P4 and 4C4). For instance, 99 can be expressed as

The methods are clever, but not entirely false. What perturbs some people is that these novel methods simplify the puzzle, which takes the fun out of it, like playing basketball with a midget hoop. Also, if these tricks weren’t enough, mathematicians devised a formula that enabled them to express numbers to infinity. Dirac is rumored to have arrived at the same generalization when he confronted the puzzle, and was thus accused of *hacking *it.

## The Hack

The hack relies on one of the most common but least understood operations – logarithm. Let me first remind you how logarithms work. If

Now, the rules demand that we only use the number 4, so the base of logarithms we employ must be limited to 4. For instance, if

Now, let’s perform the same logarithmic operation X, but replace 4 with ½. For instance,

The numbers on the right side of the equal sign increment each time we increment the number ½ is raised to. This incrementation could reach infinity if we were to increment the power on ½ *ad infinitum. *

If we express the operation on the left with four fours, we would solve or “hack” the puzzle.

Refer to X again and you will realize that the powers of

.

.

.

If exponentials weren’t enough, adroit mathematicians used logarithms to solve it. This generalization or formula to produce any natural number essentially vanquishes the puzzle once and for all. To produce any number, one must plug in the formula, the square root of 4 taken for the same number of times. Given a page long enough, one billion or even infinity can be produced by taking the square root of 4 a billion or an infinite number of times. For this reason, logarithms have been banned as a viable solution to solve the puzzle. However, some mathematicians also despise this hack because it *looks ugly*.

The puzzle has been used for years by puzzle lovers to pass the time during long journeys on trains or flights. Now, even newer versions of the puzzle have been developed, versions that might require superior mathematical acuity – the five fives or six sixes puzzles! But do you have what it takes?

#### References

**About the Author**:

Akash Peshin is an Electronic Engineer from the University of Mumbai, India and a science writer at ScienceABC. Enamored with science ever since discovering a picture book about Saturn at the age of 7, he believes that what fundamentally fuels this passion is his curiosity and appetite for wonder.

**More from this author**.

I never approved of the use of the square roots in this puzzle. It feels like cheating because the idea is to only use four 4’s and no other numbers, but by using the “square” root you’re sneaking in an invisible ”2”. And as you can see using logarithms and square roots provides a quick hack. Things the 4th root or the .4th root are fine. But we don’t typically write the two using the root symbol as by itself unless a different root is specified it is 2 by default, but the two is still there.