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Four Fours

Problem

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

For example, 1 =
4
4
+ 4 – 4, 2 =
4
4
+
4
4
and 3 =
4 + 4 + 4
4

Solution

This problem is ambiguous in its wording, "simple rules of arithmetic"; that is, are we to interpret it as only using +, –, times, 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 radical4 = 41/2 and 4! = 4 times 3 times 2 times 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 + radical4
2 =
4
4
+
4
4
15 = 4 times 4 –
4
4
3 =
4 + 4 + 4
4
16 = 4 times 4 + (4 – 4)
4 = 4 + 4(4 – 4)
17 = 4 times 4 +
4
4
5 =
4 times 4 + 4
4
18 = 4 times 4 + 4 – radical4
6 =
4 + 4
4
+ 4
19 = 4! – 4 –
4
4
7 = 4 + 4 –
4
4
20 = 4 times 4 + radical4 + radical4
8 = 4 + 4 + (4 – 4)
21 = 4! – 4 +
4
4
9 = 4 + 4 +
4
4
22 = 4 times 4 + 4 + radical4
10 = 4 + 4 +
4
radical4
23 = 4! – radical4 +
4
4
11 =
4!
radical4
4
4
24 = 4 times 4 + 4 + 4
12 = 4 + 4 + radical4 +radical4
25 = 4! + radical4 –
4
4
13 =
4!
radical4
+
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, [radical(radical4)] = [radical2] = [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 (+, –, times, 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,

radical4 = 41/2
radicalradical4 = (41/2)1/2 = 41/4
radicalradicalradical4 = ((41/2)1/2)1/2 = 41/8 and so on.

By definition,

log4(41) = 1
log4(42) = 2
log4(43) = 3, leading to log4(4x) = x.

Therefore,

log4(radical4) = 1/2
log4(radicalradical4) = 1/4
log4(radicalradicalradical4) = 1/8 and so on.

In the same way,

log1/2(1/2) = 1
log1/2(1/4) = 2
log1/2(1/8) = 3, ...

By writing log1/2(x) as log(radical4/4)(x) we can now produce any finite integer using four fours.

log(radical4/4)(log4(radical4)) = 1
log(radical4/4)(log4(radicalradical4)) = 2
log(radical4/4)(log4(radicalradicalradical4)) = 3, ...

Problem ID: 14 (Sep 2000)     Difficulty: 1 Star

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