# Combine 3, 3, 5 & 7 to get 24

Using operators plus, minus, multiplied by and divide by, and as many brackets as you want, can you do a formula which uses 3, 3, 5 and 7 to make 24?

Each number must be used and can only be used once (so there will be two 3's).

So, for example, (3x7)+3 makes 24, but this isn't valid because the 5 wasn't used.

I don't know if this is possible, btw! It's in a game I'm playing.

• Well, I noticed that $${\large 57-33=24}.$$ Does this break any rules? Jul 26, 2018 at 10:33
• It's called 24 game for anyone wondering. Jul 26, 2018 at 15:43
• @user477343 Accepted comment.
– m4n0
Jul 26, 2018 at 18:13
• Challenge 24. Nostalgia. Jul 26, 2018 at 20:58
• @user477343 Definitely a good answer puzzling.stackexchange.com/questions/5555/… Jul 27, 2018 at 6:26

Well, I tried to solve it the hard way (using double fractions) but actually it's quite easy.

(3 * 5 - 7) * 3 = (15 - 7) * 3 = 8 * 3 = 24

• Indeed, by brute-force search, this is the only formula up to commutative transformation. Jul 26, 2018 at 12:15

(3 XOR 5) x (3 XOR 7)


... yes, yes, I know XOR's not allowed. Poor XOR. Nobody ever invites him.

• Nice twist in this answer! Jul 26, 2018 at 16:32
• Could someone please explain this answer? I know XOR as a logic gate but I don't understand how can it be used here? Jul 6, 2020 at 12:53
• @defectedWBC - XOR on numbers treats each bit independently. 3 XOR 5 is: 0011 XOR 0101, which is 0110, or 6. Likewise, 3 XOR 7 is: 0011 XOR 0111 = 0100, or 4. If you're on a windows machine, you can go into Calculator and set it to Programmer mode, and perform XOR's as well. Jul 6, 2020 at 13:03

using factorials of 3
$7+5+3!+3! = 24$

• Nice but the question lists specific operators. The factorial is not one of them. I solved one of these using binomials once. Although this does not use new operators it does rely on the placement of the numbers and should not be accepted. Jul 26, 2018 at 12:44

We can use as many brackets as we want, and Wikipedia states

Square brackets, as in [π] = 3, are sometimes used to denote the floor function, which rounds a real number down to the next integer.

So I will use square brackets [] to denote the floor function.

$$([5 \div 3]+7) \times 3 = (1+7) \times 3 = 8 \times 3 = 24$$

If we massage the rules a little more, and can put numbers in different positions, we can use Falling Factorial notation, where

$$(x)_n = x(x-1)(x-2)\ \cdot \cdot \cdot (x-n+1) = \frac{x!}{(x-n)!}$$

$$[(5)_3 \div 7] \times 3 = \big[\frac{5!}{(5-3)!} \div 7 \big] \times 3 = [60 \div 7] \times 3 = 8 \times 3 = 24$$

Unfortunately, there is no way to use combinations to get the given input to 24. 🙁

• argh, it was going so well. Aug 6, 2018 at 7:45

When we

combine 3 and 3 it makes 33

and when we

combine 5 and 7 it makes 57

then

subtract 57-33

it will give the answer 24

3 + 3 + 5 + 7 + 6 = 24

Explanation:

Each number must be used and can only be used once, but the rules don't say other numbers cannot be used!

• Hmm, I guess i did miss an "only" there. Aug 6, 2018 at 7:46

$3 * (7 - 5) ^ 3$

Breaks the rule but hey

$(3*3*5)-(7*3) = 45-21 = 24$

• It's got an extra 3, so it's breaking the rules. Jul 26, 2018 at 12:59

Seems pretty straight forward, sorry I don't know how to hide it:

(5x7) - (3x3) = 35 - 9 = 24

• Sorry, this is wrong: 35 - 9 = 26, not 24. Jul 26, 2018 at 17:00
• Sure, engineer, I can do math! Out of practice it seems. Jul 26, 2018 at 17:03

check it 3!+3!+5+7 =6+6+12 =24

• Welcome to Puzzling! This uses an operator (!) which is not allowed according to the rules in the question. Jul 30, 2018 at 6:52