mathschallenge.net logo

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 (+, –, times, ÷, and parentheses), express all of the integers from 1 to 25.

For example, 1 = 2 times 3 – (1 + 4)


Solution

Of course, there are may be other ways of arriving at each of these numbers:

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

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 (+, –, times, 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,

radical2 = 21/2
radicalradical2 = (21/2)1/2 = 21/4
radicalradicalradical2 = ((21/2)1/2)1/2 = 21/8, and so on.

Therefore,

log2(radical2) = 1/2 = (1/2)1
log2(radicalradical2) = 1/4 = (1/2)2
log2(radicalradicalradical2) = 1/8 = (1/2)3, et cetera.

Hence,

log1/2(log2(radical2)) = 1
log1/2(log2(radicalradical2)) = 2
log1/2(log2(radicalradicalradical2)) = 3, ...

By using the integer part function, 1/[radical(3!)] = 1/[2.449...] = 1/2, we can obtain the required base 1/2, and using radical4 to obtain the base 2, we can now produce any finite integer using the digits 1, 2, 3, and 4.

log(1/[radical(3!)])(logradical4(radical2)) = 1
log(1/[radical(3!)])(logradical4(radicalradical2)) = 2
log(1/[radical(3!)])(logradical4(radicalradicalradical2)) = 3, ...

Problem ID: 13 (Sep 2000)     Difficulty: 1 Star

Only Show Problem