The Game of Four 4s

1.1 The Fourth Ranked King

Perhaps, nothing can give you as much fun and experience of ordinary arithmetic operations as the game of four 4s.

(Professor and eminent scientist) Jayant Narlikar, had come across it when he was in middle school. It was introduced through the following story:

A king was proud of his vast empire, and thought that his was the top-ranking kingdom in the world. He asked the scholars in his court to verify it through a world wide search. In those (pre-internet) days of the past, people had to physically travel to obtain information. And, so  the wise men travelled to the East, to the West, to the North and the South. They returned in due course with the following tidings:

“Your Majesty! Yours is the fourth largest empire on the Earth.”

The  King was disappointed and also furious. He was about to issue commands to behead these bearers of unwelcome tidings, when his Vizier stepped in.

“Sir, it is indeed a happy circumstance that you are ranked Number Four”, he added. “For, if I may be permitted to say so, of the ten primary digits 4 is the most versatile so  far as arithmetical operations are concerned.”

“Explain yourself!”, snapped the king.

The King, whatever his other shortcomings were, was well versed in elementary arithmetic. He would not be easily fobbed off.

So, the Vizier showed him the following game, a game which seemed very simple at first, but grew progressively more involved and interesting.

See for yourself, how far you can progress in the game of four 4s.

1.2 Rules of the Game

The rules of this game are simple. You are to use the number 4, four times in the well established operations of arithmetic. And you are required to construct integers 1, 2, 3, …etc.

What are the permitted operations?

(1) You can, of course, use the four fundamental operations of addition, subtraction, multiplication and division. Thus, you can express zero, one, two and three, as:


\frac{4}{4} \times \frac{4}{4}=1

\frac{4}{4} + \frac{4}{4}=2


II) Next, you can use the square root sign \sqrt{}. The fact that 4 is  a perfect square, helps, of course. Thus, you can construct \sqrt {4}=2, which may prove useful. The square root sign can cover big expression too, for example, \sqrt {4+4+\frac{4}{4}}=3.

Likewise, one can raise expressions to powers, example, 4^{4}=256

III) The next important operation is of decimalization, both regular and recurring. For example, the following expressions can be useful in constructing some numbers:



IV) A very useful permitted operation is the “factorial”. In general, for any integer N, we write

N! = 1 \times 2 \times 3 \ldots \times N. Therefore,

4! = 1 \times 2 \times 3 \times 4=24.

That is, about all! In other words, using all these operations on the number 4, use four and only four times, using no other number or symbol, but using parentheses as required, how  far can you go, starting with 1?

1.3 Examples

Just to see, how these operations work, let us try out a few examples:

1.3.1 The number 13

To make 13, we can proceed in many ways. For example,

13 = \frac{4!}{\sqrt{4}}+\frac{4}{4}

or 13 = 4 + 4 + \frac{\sqrt{4}}{4}

And, of course, there are other ways. Some are more elegant than others, but as long as an expression is mathematically correct, that is all that matters. You should have no difficulty in reaching 100 or going well past it. See, for example, a way of making a number like 87 next.

1.3.2 The number 87

This may appear difficult at first !! But, practice with smaller numbers will show the way:

87=4 \times 4! - \frac{4}{.\overline{4}}


.\overline{4} means .4 recurring decimal.

Of course, there will come a stage when you are unable to make the next number with four 4s. The larger this number is the better is your score.

1.4 In the end…

The Vizier got the King interested in this game to such an extent that the latter forgot his dismay at being ranked Number 4. And, of course, instead of being punished, the Wise Men were rewarded for their tidings.

1.5 Some problems for you!

Now that you are yourself hooked onto this game, try the following problems:

  1. Construct the following numbers using four 4s:
    (a) 516 (b) 641 (c) 3634 (d) 2187
  2. Instead of four 4s, try working with three 3s. or  five 5s, to see how much better the game was with four 4s.
  3. What is the largest number that you can construct with four 4s?

So, are you having fun now? Let me know how far you reached, 

Nalin Pithwa




One Comment

  1. Posted June 12, 2016 at 2:38 pm | Permalink | Reply

    I read this book by Narlikar in high-school, I was really fascinated. You must this book also:

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