How do you get 23 using the numbers 1, 2, 3 and 5?

Question

You can only use Addition, Subtraction, Multiplication, Division. This is for a math project for my daughter.

You can only use the numbers once and all numbers do not need to be used.

I don''t believe there is and answer to the question as stated. — paparazzo, Nov 17 '17 at 20:37
Can you use concatenation? In other words, can you make a two digit number by putting two of the digits together? Like $1$ and $5$ could make the number $15$? — Todd Wilcox, Nov 18 '17 at 12:54
Looks like this puzzle is either unsolvable or incomplete. If you wanted an unsolvability proof, it would have been better to mention that in the puzzle. — Bass, Nov 18 '17 at 14:45
Can I just use an eraser on the 1, the 5, the commas, and the word and? — Mazura, Nov 18 '17 at 17:35
@SarahFritz Please tell us the average age of your daughter classroom (7 years old?, 17 years old?) and what was the teacher expecting for this math project? "impossible", "usage of concatenation", "usage of decimal point", "usage of exclamation point" (factorial), "usage of power notation", "usage of non-decimal base", etc.? Or was it just a typo from teacher? — Cœur, Nov 19 '17 at 10:23
It's worth bearing in mind that setting a child a maths question that cannot be solved - without first introducing the concept of insoluble problems - may reinforce any feeling in the child that they are unable to answer maths questions and are therefore bad at maths. Such questions should be handled carefully to ensure that the discovery that there is no solution is a positive outcome for the child. — Vince O'Sullivan, Nov 19 '17 at 11:03
@VinceO'Sullivan, that is such a good comment, it deserves to be its own post, on parenting or something. — NH., Nov 21 '17 at 18:30

Dr Xorile · Answer 1 · 2017-11-21T18:36:38.400

42

As stated the problem is not possible. Here's an online solver to show that.

Lateral thinking options could fix it (like @Apep (reinterpretation of the list), @jlars62(decimal point (very clever)), or @hoffmale (factorials), or @sousben and @D Krueger (non-decimal)). Or allowing powers:

$5^2-3+1=23$

Or allowing concatenation.

edited Nov 21 '17 at 18:36

answered Nov 17 '17 at 20:13

Dr Xorile

23,406
3
47
125

Problem statement does not include exponentiation but still +1 – paparazzo Nov 17 '17 at 20:34
1

Powers are not allowed: -1 No solution: +1 – Elements In Space Nov 17 '17 at 22:41
13

@ElementsinSpace: powers were used as "lateral thinking" after explaining there is no solution. – Nov 18 '17 at 12:46
5² is just equal to 5*5. Wouldn't this be a solution? – SilverWolf Nov 19 '17 at 14:48
1

Per original question: You can only use each number once – Dr Xorile Nov 19 '17 at 14:50
1

This is what I arrived at independently. Using each number once. 5 squared uses 5 and 2. Squaring a number is multiplication. How semantic is the question? +1 - like the end of the calculation!! – Tim Nov 20 '17 at 09:25
1

You are, however, making an assumption (usually a good one) that the 23 stated in the title is base 10. https://puzzling.stackexchange.com/a/57085/37225 and https://puzzling.stackexchange.com/a/57082/37225 cleverly answer the question you "prove" is unanswerable. Every fact has assumptions underlying it. – NH. Nov 21 '17 at 18:32
I'll add those in. Very good. – Dr Xorile Nov 21 '17 at 18:35

score 37 · Answer 2 · answered Nov 17 '17 at 20:09

37

One solution could be:

(5+3)*3-1

under the (possibly invalid) assumption that

the problem could be considered as using "1, two 3, and 5"

answered Nov 17 '17 at 20:09

Apep

4,619
1
20
39

15

Wins the "lateral thinking" award... – smci Nov 18 '17 at 01:34

Seyed · Answer 3 · 2017-11-22T01:05:19.307

28

I guess we are not allowed to repeat the numbers:

35 - 12 = 23

edited Nov 22 '17 at 01:05

answered Nov 17 '17 at 18:58

Seyed

2,234
1
13
17

1

This has to be the simplest possible answer. :) – Sid Nov 17 '17 at 18:59
21

concatenation is not allowed as per the OP, if it was , surely just concatenate the 2 and 3=> 23 – Jason V Nov 17 '17 at 19:09
4

@JasonV, OP didn't say anything about that. And you can't concatenate the 2 and 3 because you must use all those four numbers. – Seyed Nov 17 '17 at 19:12
8

when OP said "you can only use Addition, Subtraction, Multiplication, Division" they did not include concatenation. Therefore, this is out of scope. Also, OP did not say you must use all numbers nor only once. – Jason V Nov 17 '17 at 19:14
2

@JasonV, I am sure you know that concatenating is not a part of mathematical operation. I think it is better wait and see what the OP thinks about these answers and we shouldn't talk on his behalf. – Seyed Nov 17 '17 at 19:19
1

@Seyed: Actually, concatenating is a valid and well defined mathematical operation. It's just not one of the operations allowed by the OP. (if you are wondering what mathematical, rather than string, operation concat is it is x*10+y) – slebetman Nov 19 '17 at 19:01
@slebetman, We all know that with those conditions there is no solution to this problem and in my opinion this answer is the closest to an acceptable solution. Other solutions with exponents and repeating numbers got more reputations than my solution :-) – Seyed Nov 19 '17 at 20:09
The fact that the question specifies 'numbers' and not 'digits' invalidates this answer. There is no listed operator which combines the integer numbers 3 and 5 to equal 35, likewise with 1 and 2 equaling 12. – navigator_ Dec 11 '17 at 22:58

jlars62 · Answer 4 · 2017-11-18T00:48:31.100

25

$$ \frac{5}{.2} - 3 + 1 = 23 $$

edited Nov 18 '17 at 00:48

answered Nov 17 '17 at 22:01

jlars62

351
2
6

Please put answers in spoiler tags. – Nov 18 '17 at 00:22
Doesn't .2 mean dividing 2 by 10? – DrD Nov 19 '17 at 13:25
3

@DEEM : Just because $0.2 = 2/10$ does not mean "$.2$" entails division ... any more than $4 = 8/2$ means that "$4$" entails division. – Eric Towers Nov 19 '17 at 23:49
5

Brilliant, but no. Had the question said "digits" this would work, but the question says "numbers" and the number .2 is not the same as the number 2. – Iron Pillow Nov 20 '17 at 06:07

score 19 · Answer 5 · answered Nov 17 '17 at 21:38

19

23, not using 1 or 5.

didn't even need to use any mathematical functions

answered Nov 17 '17 at 21:38

SeanC

789
4
13

1

Specifically "2" + "3" – Ambo100 Nov 18 '17 at 12:45
-1. You used an invalid operation (concatenation) that wasn't allowed in the post. – NH. Nov 21 '17 at 18:20

D Krueger · Answer 6 · 2017-11-19T14:56:21.083

13

It's very straightforward:

(5*3+2)/1

Or, as pointed out by Cœur, since all numbers need not be used:5*3+2

Why this works:

Calculations are performed in base-7.

edited Nov 19 '17 at 14:56

answered Nov 18 '17 at 19:38

D Krueger

798
1
6
10

You can remove the /1 as not all symbols are required. – Cœur Nov 19 '17 at 10:15

score 8 · Answer 7 · answered Nov 18 '17 at 21:28

8

This is the answer, only using 2 of the 4 proposed numbers:

5 * 3 = 23

How come, you say?

we used base 6 calculations

answered Nov 18 '17 at 21:28

sousben

2,586
1
12
29

score 7 · Answer 8 · answered Nov 17 '17 at 22:23

7

Using concatenation:

25 - 3 + 1

answered Nov 17 '17 at 22:23

dstudeba

179
2

score 3 · Answer 9 · answered Nov 17 '17 at 20:52

3

Using numbers more than once, but interesting sequence.

1*2+2*3+3*5 = 23

answered Nov 17 '17 at 20:52

navigator_

131
1

Given that the question is unclear as to if we can use each number more than once, this answer is plausible... – Jason V Nov 17 '17 at 20:55
fibonacci new world order confirmed? – MCMastery Nov 19 '17 at 02:27

hoffmale · Answer 10 · 2017-11-18T18:37:59.253

3

If factorial is allowed:

(5 - 1)! - 3 + 2 = 23

or

5 * (3! - 1) - 2 = 23

or

(2 + 1) * 3! + 5 = 23

or

5! / 3! + 2 + 1

edited Nov 18 '17 at 18:37

answered Nov 18 '17 at 06:58

hoffmale

277
2
6

What's faculty? – Mazura Nov 18 '17 at 17:41
I think hoffmale means factorial, or at least that's what I would call the ! operator – Foon Nov 18 '17 at 18:35
@Mazura yeah, i meant factorial... bad/wrong translation from german – hoffmale Nov 18 '17 at 18:36
5

"factorial" is "Fakultät" in German, which is basically the same word used for an university "faculty" – hoffmale Nov 18 '17 at 18:43
factorial was the first solution that came to my mind! +1 – Sarthak Mittal Nov 20 '17 at 06:56

score 3 · Answer 11 · edited Nov 18 '17 at 14:20

3

Assuming concatenation is allowed then this is another answer:

13 + 2 * 5

edited Nov 18 '17 at 14:20

Todd Wilcox

538
4
11

answered Nov 18 '17 at 12:26

David Yaw

357
1
4

score 3 · Answer 12 · answered Nov 21 '17 at 07:40

3

How about this

(3*2+1)*5 = 23 using HEX

answered Nov 21 '17 at 07:40

SJFJ

181
3

1

HEX does not sound like addition, subtraction, multiplication or division. – boboquack Nov 21 '17 at 08:29
@boboquack Why can't I use hexadecimal, where the base is 16? – SJFJ Nov 21 '17 at 08:40
Base conversion is a way around literally any of these puzzles, and gets quite boring if done over and over again (a couple of samples). It also doesn't answer the intended question, even if it does answer the literal question. Lastly, I argue that base 16 requires $(3\cdot2+1)\cdot5=23_{16}$, which requires an extra 16. – boboquack Nov 21 '17 at 08:53
@boboquack Ok, new to the site so I wasn't aware that it was considered boring. – SJFJ Nov 21 '17 at 09:09
1

@boboquack, boring or not, the question had better state that the number they want is $23_{10}$ if they are trying to rule out creative solutions. – NH. Nov 21 '17 at 18:35

score 3 · Answer 13 · answered Nov 21 '17 at 08:38

3

Dr Xorile has determined that there is no solution to the problem as stated. So all that remains are out-of-the-box solutions. Some good approaches have already been presented, treating "+" as string concatenation, and changing bases among the best.

Here's what might be jokingly termed a statistician's approach:

You could attempt the question twice and take the average:

5(3+1)+2 = 22
5(3+2)-1 = 24

Average = $\frac{22+24}{2}$ = 23.

All conditions are fulfilled on each attempt. :D

answered Nov 21 '17 at 08:38

Lawrence

7,919
2
22
56

so fuzzy math ? – NH. Nov 21 '17 at 18:34
@NH. All crisp. Stochastic, maybe. :) – Lawrence Nov 21 '17 at 22:44

score 2 · Answer 14 · answered Nov 17 '17 at 18:54

2

If we can use a number twice, then:

(2+2)*5+3 = 23.
OR: (2+2)*3*2*1-5+(2*2)

answered Nov 17 '17 at 18:54

Sid

15,153
2
43
94

2

in that case, how about 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1 = 23? – Daniel Vestøl Nov 20 '17 at 14:52
@DanielVestøl My answer was before the OP edited the question – Sid Nov 20 '17 at 15:21

score 2 · Answer 15 · answered Nov 17 '17 at 20:41

2

(5^2)-3+1

Squaring a number is the same as multiplying it by itself, so this counts in my book.

answered Nov 17 '17 at 20:41

Guest1234

39
1

3

I don't think that was the OP's intent. All other operations can be reduced to simple addition, and by breaking it down to this you are saying 5*5 which gives two 5s. If the OP says we can use the number twice, this works. – Jason V Nov 17 '17 at 20:53
2

Basically the same as @DrXorile's answer. – Tom Carpenter Nov 17 '17 at 21:12

score 2 · Answer 16 · answered Nov 18 '17 at 06:28

2

Perhaps,

Round of (51/2) - 3

That is

26 - 3 to fetch 23

Of course, this involves concatenation.

answered Nov 18 '17 at 06:28

Mea Culpa Nay

8,626
1
13
45

directly "23" is simpler – Cœur Nov 19 '17 at 10:25

How do you get 23 using the numbers 1, 2, 3 and 5?

16 Answers16