One 4, a story

Colin Backhurst,
23 Mar 2000

I used to play the Four 4's game many years ago with my friends. Of course there is always the problem of which operations to allow. We all accepted that the basic arithmetical operations should be allowed (+,-,*,/), and we also allowed powers, roots, decimal points, and factorials. Other things were a matter of debate.

In the end we decided that we would allow anything that was on our calculators, and so this also brought in reciprocals, logs and trig functions. This didn't appear to give us anything new, as using these functions usually led to a decimal number (and taking its integer part would be considered to be "cheating" as there wasn't a calculator function to do this).

I spent some weeks steadily setting higher and higher records until I realised that

reciprocal(cos(arctan(4))) = 17
reciprocal(cos(arctan(reciprocal(cos(arctan(4)))))) = 18

Continuing this process another seven times I reached sqrt(25) which is 5, and eleven steps further 6. Continuing on you will eventually reach any integer larger than 4.


tan(arccos(recip(4))) = 15

and so repeating this you can also reach the numbers less than 4 as well.

So using just one 4 (or even one "n"), I had a formula for any other number. Needless to say this was the end of the Four 4's game.

Related material

  • Funny Arithmetic
  • A single formula to express all numbers
  • Any Integer with Three 2s
  • Representation of numbers with a single 4. The rules and the possibilities
  • Three 3's
  • Representation of numbers with three 4's
  • Representation of numbers with three 5's
  • Representation of numbers with four 3's
  • Representation of numbers with four 4's
  • Representation of numbers with four 5's
  • A problem of representing 4 by four identical digits
  • A problem of representing 6 by three identical digits
  • A 9's Fan's Clock
  • Make an Identity
  • Fun with numbers: place plus/minus signs between the digits
  • 24 with One Digit
  • Representation of 3 with Three Equal Digits
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