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Given the sequence 123456789:

You can insert three operations ($+$,$-$,$\times$,$/$) into this sequence to make the equation = 100.

My question is: is there a way to solve this without brute force?

(I tried to represent it as a graph but I'm unsure where to go from there.)

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    As usual with this kind of question, details are missing. Are parenthesis allowed ? How does operator precedence work ? Are inexact divisions allowed ? Are intermediate fractional values allowed ?2015-03-19

1 Answers 1

11

The strategy I would use is the following:

Goal is to get $100$ as sum. Among the digits $123456789$, pick and choose the sum close to $100$, such as $89$. Therefore I would attempt to get a value of $11$ from $1234567$ using different combinations.

When you start working on a smaller sum now (sort of like divide and conquer), you may get the desired result. (Of course there is no specific algorithm).

In order to get $11$, I have

$(1\times 23)-4+5-6-7 = 11$

$(1-2+3-4+5)\times 6 -7= 11$

$123-45-67 = 11$

Therefore

$(1\times 23)-4+5-6-7+89 = 100$

$(1-2+3-4+5)\times 6 - 7+89=100$

$123-45-67+89=100$

${\bf{Adding}}$ ${\bf{more}}$ to it: If we look at $78+9 = 87$ and instead of $89$, we seek the remaining $13$ to be derived from $123456$, and one way to get that is

$6+5+4-3+2-1=13$

Therefore

$78+9+6+5+4-3+2-1=100$