
As Easy As 1234
Problem
Using each of the digits 1, 2, 3, and 4, once and only once, with the basic rules of arithmetic (+, , , ÷, and parentheses), express all of the integers from 1 to 25.
For example, 1 = 2 3 (1 + 4)
Solution
Of course, there are may be other ways of arriving at each of these numbers:
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Extensions
- If you are now permitted to use square roots, exponents, and factorials, can you produce all of the integers from 1 to 100?
- 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 ÷)?
- What is the largest known prime you can produce?
Notes
Surprisingly it is possible to produce any finite integer using logarithms in a rather ingenious way.
We can see that,






Therefore,
log2(
2) = 1/2 = (1/2)1
log2(
2) = 1/4 = (1/2)2
log2(

2) = 1/8 = (1/2)3, et cetera.

log2(


log2(



Hence,
log1/2(log2(
2)) = 1
log1/2(log2(
2)) = 2
log1/2(log2(

2)) = 3, ...

log1/2(log2(


log1/2(log2(



By using the integer part function, 1/[(3!)] = 1/[2.449...] = 1/2, we can obtain the required base 1/2, and using
4 to obtain the base 2, we can now produce any finite integer using the digits 1, 2, 3, and 4.
log(1/[(3!)])(log
4(
2)) = 1
log(1/[(3!)])(log
4(
2)) = 2
log(1/[(3!)])(log
4(
2)) = 3, ...
Problem ID: 13 (Sep 2000) Difficulty: 1 Star