# Use digits 3 3 5 to make 67 or 97

Rules:

1. All 3 digits — $$3$$, $$3$$, and $$5$$ — must be used once each in any order. You can concatenate these digits to create multi-digit numbers (i.e. $$33$$).
2. You can use the factorial operation ($$n!$$), the subfactorial operation ($$!n$$) and the double factorial operation ($$n!!$$) (i.e. $$3!=6$$, $$!5=44$$, $$(3!)!!=48$$). However, extended multi-factorials ($$n!!!...$$) cannot be used.
3. $$+$$, $$-$$, $$\times$$, $$/$$,$$()$$, $$\hat{}$$ can be used for functions.
4. You cannot use: round, floor, ceiling, truncate function or functions such as sin, cos, log.
• Is sqrt allowed? Also, what about repeating decimal operator? Also, what about decimal points, such as ".3"?
– JS1
Jul 2 '19 at 23:16
• Do you mean 67 and 97, or 67 or 97? If you mean or, consider giving @malioboro's answer a checkmark.
– Duck
Jul 2 '19 at 23:17
• @Duck that answer uses concatenation after modifying the numbers, which is not the same thing as making a multi-digit number out of the original digits.
– Bass
Jul 3 '19 at 6:34
• Same problem occurred here with my answer: puzzling.stackexchange.com/questions/84834/…
– Duck
Jul 3 '19 at 15:30

I think this works for 97:

$$!(3!)-5!-(3!)!!=97$$

• @Neil W is there any way to combine our answers? Jul 4 '19 at 7:51
• Nice find! For those wondering, this simplifies to $265 - 120 - 48 = 97$
– JS1
Jul 4 '19 at 9:03
• I don't even know how lucky I was to find this @JS1 Jul 4 '19 at 15:39

With a decimal point

$$67 = .5^{-(3!)}+3$$

• I like this much better than the one with concatenation, because with concatenation it is easy to make both 67 and 97. Example: $5! - ((!3) || 3) = 120 - (2 || 3) = 120 - 23 = 97$
– JS1
Jul 3 '19 at 17:08

for 67:

$$3! \Vert (!3+5)$$

• Wait, that's not partial, it says 67 or 97, not both. +1 :)
– Duck
Jul 2 '19 at 22:05
• wait, that is a concatenation, isnt it? is that allowed?? Jul 2 '19 at 23:31
• Doesn't say concatenation isn't allowed.
– Duck
Jul 2 '19 at 23:36
• @OmegaKrypton I'm not pretty sure, actually. I think the first rule is said that we allowed doing something like concatenation Jul 3 '19 at 0:07
• I think that is only applicable for the original numbers? Jul 3 '19 at 0:32