mathschallenge.net

Problem

Using exactly four fours and the simple rules of arithmetic, express all of the integers from 1 to 25.

Solution

This problem is ambiguous in its wording, "simple rules of arithmetic"; that is, are we to interpret it as only using +, –, , and ÷, or will we need to employ a few tricks? Unfortunately it is not possible to obtain all of the integers from 1 to 25 otherwise.

Useful building blocks are square roots and factorial, however, as 4 = 4^1/2 and 4! = 4 3 2 1, they both use numbers that we do not have. What about the use of place positions, as in 44 or .4? I personally object to this type.

Wherever possible, I have tried to use methods that are as close to the basic rules of arithmetic as possible: add, subtract, multiply and divide, submitting to the use of square root and factorial only where necessary.

1 = 4 4 + 4 – 4 14 = 4 + 4 + 4 + 4 2 = 4 4 + 4 4 15 = 4 4 – 4 4 3 = 4 + 4 + 4 4 16 = 4 4 + (4 – 4) 4 = 4 + 4(4 – 4) 17 = 4 4 + 4 4 5 = 4 4 + 4 4 18 = 4 4 + 4 – 4 6 = 4 + 4 4 + 4 19 = 4! – 4 – 4 4 7 = 4 + 4 – 4 4 20 = 4 4 + 4 + 4 8 = 4 + 4 + (4 – 4) 21 = 4! – 4 + 4 4 9 = 4 + 4 + 4 4 22 = 4 4 + 4 + 4 10 = 4 + 4 + 4 4 23 = 4! – 4 + 4 4 11 = 4! 4 – 4 4 24 = 4 4 + 4 + 4 12 = 4 + 4 + 4 +4 25 = 4! + 4 – 4 4 13 = 4! 4 + 4 4

Can you produce all of the integers from 1 to 100?

The integer part function, [x], which has the effect of stripping away the decimal fraction of a number and leaving the integer part, can be used to derive some of the more stubborn numbers. Hence, [(4)] = [2] = [1.41...] = 1. Similarly, [4cos4] = [3.99...] = 3 (working in degrees) and [log(4)] = [0.602...] = 0 (working in base 10), and this could be used to eliminate a surplus 4.

Once you"ve given the integers 1 through 100 a good shot, you may like to check out David Wheeler's rather outstanding, Definitive Four Fours Answer Key; found at: http://www.dwheeler.com/fourfours/. Wherever he employs contentious functions: for example, square(n)=n², which makes use of a 2 in the exponent, he has gone to great lengths to provide an "impurity" index admitting the use of functions which some people may object to using.

Extensions

• What is the first natural number that cannot be derived?
• Which is the first number that cannot be obtained if you are only permitted to use the basic rules of arithmetic (+, –, , and ÷)?
• The maximum integer is not a sensible question, as we could apply factorial any finite number of times. But, what is the largest known prime you can produce?
• Using three fours (or threes), which integers can you make?
• Using any number of fours and only addition, subtraction, multiplication and division, produce all of the integers from 1 to 25 in the most efficient way possible.

Notes

Surprisingly it is possible to produce any finite integer using logarithms in a rather ingenious way.

We can see that,

4 = 4^1/2
4 = (4^1/2)^1/2 = 4^1/4
4 = ((4^1/2)^1/2)^1/2 = 4^1/8 and so on.

By definition,

Therefore,

log₄(4) = 1/2
log₄(4) = 1/4
log₄(4) = 1/8 and so on.

In the same way,

By writing log_1/2(x) as log_(4/4)(x) we can now produce any finite integer using four fours.