1 2 3 4 5 6 7 8 9 = 100

  • The sequence of numbers $1\ 2\ 3\ 4\ 5\ 6\ 7\ 8\ 9$ has the property that you can insert mathematical operators in between the numbers from $1$ to $9$ and make the expression evaluate to 100. For example:

    $$1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 \times 9 = 100$$

    There are possibly hundreds of solutions to this problem, involving different varieties of operators. What is the expression with the fewest number of operators inserted (out of the set $+, -, \times, \div$ and maybe $\sqrt{}$ and $!$) that evaluates to 100?

    What is the assumption if I do not insert a symbol? Does 1 2 3 just become the number 123?

    Yes, exactly. Similarly, if you insert no symbols at all, then the number is just 123456789.

    Can I insert a symbol between = and 100?

    If it can beat the current solution, try. But generally, you can only insert symbols on the left side. (Also, no modifying the equals sign to do "not equals". That's a standard loophole and it doesn't apply.)

    Can I add digits in between the integers?

    Well anyways if I can, I have a cute solution with 4 added symbols: $1-2\times3345+6789 = 100$.

    @JoeZ. You should define the question such that you don't need to make *ex post facto* judgements on what is or isn't allowed. You failed to specify specifically which "mathematical symbols" are allowed and their meaning. Numbers are symbols, and they are used in mathematics. Taking it a step farther, I could invent a new operator with its own symbol (as is often done in mathematics) and define it to always evaluate to 100. Stick it anywhere on the left and *voila*: 100 = 100, one added symbol.

    @JoeZ. I wasn't complaining or implying that my hypothetical solution should be considered a valid solution to the puzzle, but trying to point out that if you explicitly listed the specific operators and their acceptable meanings (some operators have different or multiple meanings in different areas of mathematics) you would not need to contend with a plethora of "clever" solutions utilizing lateral thinking.

    @JoeZ. I believe my first comment was and still is quite constructive: it gives specific advice for how to describe the puzzle less ambiguously. Namely, by listing each symbol/operator which is acceptable and what meaning it has. What irritates me is when people misinterpret constructive criticism and cry foul instead.

    @bcrist I apologize. I got antsy about this because I've been accused of moving the goalposts in much the same way at PCG.SE (the code golf site) more than once, and it pattern-matches to people complaining about "chameleon questions" where people do indeed believe their loophole-abusing solutions should be considered valid even if they ruin the challenge, almost as a penalty to the question-asker for not making the question clear enough in the first place. Looking back at my first reply to you, some of my negativity was pretty uncalled for, so I've just deleted my comments.

    If it wasn't for your last edit 123456789 != 100 might have been a short good programmer-ish solution :)

    123456789 factorial is nowhere close to 100, though.

    @bcrist That being said, I think I would have been more likely to have seen your comment as constructive if there weren't all that framing about "You should define the..." etc., which triggered the pattern-matching I mentioned regarding the people at PCG.SE. I'm always happy to better define my problems, but I'm more likely to accept direct suggestions for that reason.

    @JoeZ. I'll admit; starting my first two sentences with "You should" and "You failed" probably set an overly negative tone.

    FWIW, here are all solutions using only +, -, / and *. https://gist.github.com/schleppy/6e121d7f5e55b71797dd

    I gotta say, it's a fine enough puzzle but I feel that it's getting *way* too much attention given that the accepted answer is easily found with a Google search.

    @EngineerToast I know. This is yet another chestnut, it seems.

    As the question is worded "a sequence of numbers", the only answer is 8. Since the question was not worded as "a sequence of digits", the digits of numbers have to be taken as is and cannot be clumped into NEW numbers. (That is, 1 and 2 are exactly that. They are not together, the number 12).

    @ThomasEding I specified in a comment above that numbers without operators between them would be strung together automatically, but if you want I'll edit that into the question to be *super* clear.

  • I believe that this is the smallest:

    $123 - 45 - 67 + 89 = 100$

    This is the smallest one I know of too, but let's see if someone can beat it.

    Unless something like 123456789>=100 counts, I couldn't find a better one either

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