Create 76 out of 1,2,3,4

Question

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.

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. — Parcly Taxel, Nov 13 '22 at 01:24
Possible duplicate of https://puzzling.stackexchange.com/questions/107527/calculation-the-hard-way Its a really nice puzzle to play with students to teach them how to do math in their head :) — Nurator, Nov 14 '22 at 07:14

Parcly Taxel · Accepted Answer · 2022-11-13T05:57:12.537

28

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.

edited Nov 13 '22 at 05:57

answered Nov 12 '22 at 09:45

Parcly Taxel

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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. – Minot Nov 12 '22 at 13:52
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@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. – Parcly Taxel Nov 12 '22 at 15:09
That's a very nice solution, I don't think I ever would off found that without code, onwards to 100! – Ri Ignis Nov 12 '22 at 20:55
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@RiIgnis Wow, 9.33 * 10^157 is quite the goal! – Iceman76 Nov 13 '22 at 01:19
@Iceman76 Where did that number come from? – Parcly Taxel Nov 13 '22 at 01:23
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"100 factorial" – Jasen Nov 13 '22 at 02:03

score 4 · Answer 2 · answered Nov 11 '22 at 23:10

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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

answered Nov 11 '22 at 23:10

will

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Yeah, no concatenation unfortunately though would be interesting to see how high you could get if it was allowed. – Ri Ignis Nov 12 '22 at 00:37
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If you are going to use concatenation, you may as well say 4^3 + 12, which does not require the "&" symbol ;) – wimi Nov 12 '22 at 17:08
@wimi - Good call! That might count as a legit answer. Each digit used once. No sneaky interpretation of the operators required. – will Nov 12 '22 at 19:18
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@wimi $2×(41-3)$ is simpler. – Parcly Taxel Nov 13 '22 at 04:00
Coincidentally as I read the initial question, my brain instantly delivered "4^3" and 4^3+12 ^^ – eagle275 Nov 14 '22 at 08:28

Create 76 out of 1,2,3,4

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