The trouble with integers is that we have examined only the very small ones. Maybe all the exciting stuff happens at really big numbers, ones we can’t even begin to think about in any very definite way. Our brains have evolved to get us out of the rain, find where the berries are, and keep us from getting killed. Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions.
Ronald Graham
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There is a famous mathematics problem in which we are allowed to use the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 and any mathematical operations to generate a sum 100. In this article, we are going to show the way to do this. Here, we have considered a few ways only, so the list is not exhaustive. If you came across any other way(s) to represent a number, not listed here, please do get back to us.
On other hand, among many mathematical operations, addition (+), subtraction (−), multiplication (×), division (÷), fraction (a/b), exponentiation (ax), decimal (.) and recurring decimal etc. are well known. We are going to use a few of them at a time.
The original question can be broken down into the following sub-questions. Our first such question is
How to write 100 using the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 only once, any mathematical operation(s) and preserving the usual ordering of the digits?
Here are few possible solutions.
100 = 1 + 2 + 3 – 4 + 5 + 6 + 78 + 9
100 = 1 + 2 + 34 – 5 + 67 – 8 + 9
100 = 1 + 23 − 4 + 56 + 7 + 8 + 9
100 = 1 + 23 – 4 + 5 + 6 + 78 – 9
100 = 12 + 3 – 4 + 5 + 67 + 8 + 9
100 = 12 + 3 + 4 + 5 – 6 – 7 + 89
100 = 12 – 3 – 4 + 5 – 6 + 7 + 89
100 = 123 – 4 – 5 – 6 – 7 + 8 − 9
100 = 123 + 45 – 67 + 8 − 9
100 = 123 + 4 – 5 + 67 − 89
100 = 123 − 45 − 67 + 89
100 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 × 9
100 = 1 × 2 × (3 + 4) × 5 + 6 + 7 + 8 + 9
100 = – 1 + 2 – 3 + 4 + 5 + 6 + 78 + 9
100 = – 1 × (2 + 3 + 4 + 5) + (6 × 7) + (8 × 9)
100 = 1 × 2 × 3 – 4 × 5 + 6 × 7 + 8 × 9
Now the second question is
How to make 100 using the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 only once and any mathematical operation(s)?
The digits may not come in the usual sequence. Here are few possible solutions.
100 = 1 + 2 + 43 + (5 × 6) + 7 + 8 + 9
100 = 1 × 2 × 4 – (3 + 6) + 5 × 9 + 7 × 8
100 = (1 + 9) × 2 × 5 + 3 + 4 + 7 – (6 + 8)
100 = (1 + 2 + 8 + 9) × 5 + 3 – (6 + 4) + 7
100 = 1 × (6 + 8) + (7 × 5 – 2) × 3 – (9 + 4)
100 = (1 ÷ 2) × 6 × 3 + (7 + 5) × 8 + 4 – 9
100 = 1 + (3 + 8) × 9 + 5 + 7 – (6 + 4 + 2)
100 = 2 + 3 + 4 + 5 – 6 – 7 + 8 + 91
100 = 2 × 5 + 37 + 64 + 8 – 19
100 = 4 × 5 × 6 – (9 ÷ 3) – 7 – 1 × (2 + 8)
100 = 9 × 8 + 6 × 5 – 2 + 3 + 4 – 1 × 7
The third question goes like this
How to write 100 using fractional numbers formed by the use of digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 only once and any mathematical symbol(s)? Here are few possible solutions.
Our next question is
How to write 100 as the sum of few terms (including exponential) formed by the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 only once?
Here are few possible solutions.
100 = 1 + (23) + (4 × 5) + 6 + (7 × 8) + 9
100 = 153 + 2 + 4 + 6 + 8 + 79
100 = (1 + 9)4 – 2 + 3 + 5 + (6 – 7) × 8
100 = 1 + 36 − 2 + 4 + 5 + (8 – 7) × 9
Finally, we can do the same task if we allow the recurring decimal. One possible solution is
Your suggestions are eagerly and respectfully welcome! See you soon with a new mathematics blog that you and I call “Math1089 – Mathematics for All!“.