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You can use +,-,*,/,^,! and sqrt operations (no concatenation) and all numbers must be used once and only once. for example, 75 can be made by $3×(4!+2-1)$ and $81=(2-1)×3^4$

I've been trying to reach 100 and this appears to be the first impossible number. Similar using 1 to 5, 202 appears to be the first impossible number.

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    $\begingroup$ For your comment about 202, you can do $202=(5*3^4-1)/2$ $\endgroup$
    – ThomasL
    Commented Nov 12, 2022 at 19:30
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    $\begingroup$ A refined version of my program gave solutions for the 1–5 problem for all but 6 numbers in 0–1000: 892, 917, 919, 926, 941, 942. $\endgroup$ Commented Nov 13, 2022 at 1:24
  • $\begingroup$ Possible duplicate of puzzling.stackexchange.com/questions/107527/… Its a really nice puzzle to play with students to teach them how to do math in their head :) $\endgroup$
    – Nurator
    Commented Nov 14, 2022 at 7:14

2 Answers 2

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I wrote a program and it returned this: $$76=4×\sqrt{(3!)!/2+1}$$

The output (using reverse Polish notation) of a refined version of my program, capable of handling the 1–5 problem, can be found here. Only 85, 86 and 93 are impossible for the 1–4 problem from 0 to 100, while 892, 917, 919, 926, 941 and 942 are impossible for the 1–5 problem from 0 to 1000.

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    $\begingroup$ Can you briefly explain how the script operates? I am very interested in it but it's very hard to read from the code as it is. $\endgroup$
    – Minot
    Commented Nov 12, 2022 at 13:52
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    $\begingroup$ @Minot I typed the whole script on my smartphone. It is basically a depth-limited search over expressions with respect to the number of operators. $\endgroup$ Commented Nov 12, 2022 at 15:09
  • $\begingroup$ That's a very nice solution, I don't think I ever would off found that without code, onwards to 100! $\endgroup$
    – Ri Ignis
    Commented Nov 12, 2022 at 20:55
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    $\begingroup$ @RiIgnis Wow, 9.33 * 10^157 is quite the goal! $\endgroup$
    – Iceman76
    Commented Nov 13, 2022 at 1:19
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    $\begingroup$ "100 factorial" $\endgroup$
    – Jasen
    Commented Nov 13, 2022 at 2:03
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This is potentially not quite in the spirit of the puzzle, but how about:

we try a little concatenation? You said we can use "and".

=(1&3^2) x 4 = 76
=(1&9) x 4 = 76
=19 x 4 = 76

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  • $\begingroup$ Yeah, no concatenation unfortunately though would be interesting to see how high you could get if it was allowed. $\endgroup$
    – Ri Ignis
    Commented Nov 12, 2022 at 0:37
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    $\begingroup$ If you are going to use concatenation, you may as well say 4^3 + 12, which does not require the "&" symbol ;) $\endgroup$
    – wimi
    Commented Nov 12, 2022 at 17:08
  • $\begingroup$ @wimi - Good call! That might count as a legit answer. Each digit used once. No sneaky interpretation of the operators required. $\endgroup$
    – will
    Commented Nov 12, 2022 at 19:18
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    $\begingroup$ @wimi $2×(41-3)$ is simpler. $\endgroup$ Commented Nov 13, 2022 at 4:00
  • $\begingroup$ Coincidentally as I read the initial question, my brain instantly delivered "4^3" and 4^3+12 ^^ $\endgroup$
    – eagle275
    Commented Nov 14, 2022 at 8:28

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