Make 100 from the digits 1 to 9 in order

Question

Write the digits 1 2 3 4 5 6 7 8 9 in that order. Between some neighbouring digits insert a plus or a minus sign; where you insert nothing, the digits join to form a longer number (so 1 2 3 could become 123, or 12 + 3, or 1 + 23, and so on). Make the expression equal exactly 100. Find a solution with as few signs as possible, and one with many.

Accepted Answer

Fewest signs: 123 minus 45 minus 67 + 89 = 100. Many signs: 1 + 2 + 3 minus 4 + 5 + 6 + 78 + 9 = 100. Check the first. 123 minus 45 = 78; 78 minus 67 = 11; 11 + 89 = 100. Three signs. Check the second. 1 + 2 + 3 = 6; minus 4 gives 2; + 5 + 6 gives 13; + 78 gives 91; + 9 gives 100. Seven signs. How many solutions. With signs allowed only between digits (no leading minus), there are 3 to the 8 = 6,561 ways to choose "plus, minus or join" in the eight gaps, and exactly 11 of them equal 100, including 12 + 3 minus 4 + 5 + 67 + 8 + 9, 1 + 2 + 34 minus 5 + 67 minus 8 + 9, and 1 + 23 minus 4 + 56 + …

Make 100 from the digits 1 to 9 in order

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