3:3! It's a football score!

Question

Given to me, by a friend:

How would you make 20, using two threes?

You may use any basic operation. And others, such as square roots (the symbol), factorials, etc. Any operation is allowed.
Just two threes, though. You can't raise a three to any digit other than three, etc.

Any out of the box, non-mathematical solutions are encouraged.

PS- Wow! That's an amazing number of responses! Quite a community, here.

EDIT: @CameronWhite and @Yly have provided, perhaps the most elegant answers. And basis your votes, I'm going ahead to mark @Cameron's answer as accepted. No issues, I hope. Although, I did think @KeyboardWielder's answer was rather crafty. And @dcyfj... Well, that would have taken some time. Awesome answer. This really isn't an edit, is this?

EDIT: Perhaps I should have specified the level of out-of-the boxness allowed. Forgive my mistake. This was, my first post, and I was surprised by the number of answers. Thanks for taking the time.

I was originally thinking of it on the lines of @KeyboardWielder's answer, after defining a function went out the window. — Athena, Jun 06 '16 at 19:11
"When twenty is just six" and "How would you make 20, using two threes?" don't necessarily mean the same thing. — GRW, Jun 06 '16 at 22:55
Indeed, indeed, @JonMarkPerry, you could make 20 (currency of choice) by betting on that score — humn, Jun 07 '16 at 18:51

score 111 · Answer 1 · edited Jun 07 '16 at 03:58

111

Does this count?

$\dbinom{3!}{3} = 20$

edited Jun 07 '16 at 03:58

JMP

35,612
7
78
151

answered Jun 07 '16 at 03:55

Yly

759
1
4
7

6

I like this one the best. – Jun 07 '16 at 14:21
So simple, I love it! – Martijn Jun 07 '16 at 17:46
It is just 2! How is this right? – Gaurav Agarwal Jun 08 '16 at 18:41
4

@GauravAgarwal: Read it as "6 choose 3", which is the binomial coefficient equating to the number of different ways to choose 3 elements from a set of 6. – Michael Myers Jun 08 '16 at 19:34
@MichaelMyers Thank you. In light of this knowledge, this is truly the most elegant answer. – Gaurav Agarwal Jun 08 '16 at 21:49

score 108 · Accepted Answer · answered Jun 07 '16 at 02:36

108

If you are allowed to use decimals, then

$$\frac{3!}{.3} = 20$$

answered Jun 07 '16 at 02:36

Cameron White

516
2
4
3

score 71 · Answer 3 · edited Jun 06 '16 at 18:57

71

Since the puzzle oddly and specifically mentions the symbol for the square root, I used this:

but rotated and reflected it giving:

edited Jun 06 '16 at 18:57

JMP

35,612
7
78
151

answered Jun 06 '16 at 18:56

Dan Russell

16,028
3
45
97

1

definitely "out-of-the-box" lol – dcfyj Jun 06 '16 at 18:57
9

Man, oh man... That, is rotating the box. But a very cool observation. – Athena Jun 06 '16 at 19:01
you could do that with factorial if you just bend it a little bit and hide the dot – JMP Jun 06 '16 at 19:03
You would be abusing it. – Athena Jun 06 '16 at 19:05
1

@RipTide You did encourage "out of the box, non-mathematical"! – Dan Russell Jun 06 '16 at 19:10
The poor, poor factorial. But yes. I did ask for it. By that logic, though, any symbol can be modified enough, no? – Athena Jun 06 '16 at 19:15
Sure. I did refrain from $(3/3)*XX=20$ in Roman numerals at least. – Dan Russell Jun 06 '16 at 20:45
2

I wish I could +20 this. That note the symbol was driving me crazy. – Devsman Jun 07 '16 at 18:35
@RipTide At least write "encourage" on your post. – EKons Jun 08 '16 at 09:46

score 67 · Answer 4 · edited Jun 07 '16 at 18:04

67

Another answer could be

$\lfloor \sqrt{(3!)!}-3! \rfloor$
$= \lfloor \sqrt{6!}-3! \rfloor$
$= \lfloor \sqrt{720}-6 \rfloor$
$= \lfloor 26.8328...-6 \rfloor$
$= \lfloor 20.8328... \rfloor$
$= 20$

edited Jun 07 '16 at 18:04

KeyboardWielder

5,956
25
55

answered Jun 06 '16 at 19:21

dcfyj

7,756
23
64

That fits, as well. – Athena Jun 06 '16 at 19:25
I would've put each equation on a separate line, but for the life of me I can't seem to figure it out. – dcfyj Jun 06 '16 at 19:26
You should add parentheses between the postfix symbols; see this Wiki article. (Tried to keep the comment spolier-free). – Angew is no longer proud of SO Jun 07 '16 at 10:58
This uses approximation. Also, $\lfloor\sqrt{3!!}-3!\rfloor$ is not invalid. – EKons Jun 07 '16 at 16:17
@ΈρικΚωνσταντόπουλος: There is no mathematical approximation, it is just abbreviation in writing (filling up the page with an infinite or so number of digits would be a highly irrational thing to do). – KeyboardWielder Jun 07 '16 at 18:06
@ΈρικΚωνσταντόπουλος Like KeyboardWielder said, I was just cutting it off. I didn't much care to be writing hundreds of numbers. – dcfyj Jun 07 '16 at 19:02
@dcfyj I didn't know you cut it off, since you hadn't included $...$ like $26.8328...$. – EKons Jun 08 '16 at 09:44
@Alexander Spoiler-free? What do you mean SPOILER >!hmm, spoiler? – EKons Jun 08 '16 at 16:29

KeyboardWielder · Answer 5 · 2016-06-07T17:55:09.187

40

The simplest one is:

three + three = 20 (in base 3)

edited Jun 07 '16 at 17:55

answered Jun 06 '16 at 18:46

KeyboardWielder

5,956
25
55

Base 3 on the right. – Athena Jun 06 '16 at 19:04
@KeyboardWielder, +1 on the logical side. – Athena Jun 06 '16 at 19:07
16

This would work for all bases greater than 3, too. $x+x=20_x$. – Engineer Toast Jun 06 '16 at 19:13
1

if you are doing math base X, then X+X=10, right? or maybe i am confused by using the "3" character while claiming to be "base 3"... surely base "base 3" limits itself to the digits 0, 1, and 2? – Leo Lansford Jun 06 '16 at 20:10
@LeoLansford Not quite- x/2 + x/2 would equal 10. In base X, X=10. – Patrick N Jun 06 '16 at 20:13
11

The question title specifically says "twenty", which is a specific value, not a representation of a value in any base. three + three will always equal six, not twenty. The question does, however, says "20" – Suppen Jun 07 '16 at 06:55
4

three + three = 20 but 20 is not twenty in base 3. – user253751 Jun 07 '16 at 07:07
2

@EngineerToast That's why I've started to increase the base of my age instead of the years. I'm 21 now. Base 17. – Hendrik Wiese Jun 07 '16 at 08:27
except 3 does not exist in base 3 – njzk2 Jun 07 '16 at 14:29
3

@Suppen Number 20 is pronounced as "twenty" regardless of the base. So three + three = twenty in base three. However, the problem of this solution is that in order to be correct, it requires third three on the right side, which makes it incorrect regarding the puzzle limitations. – fbxmg Jun 07 '16 at 14:32
@njzk2 Left side is supposed to be in decimal base, only right side is in base 3. – fbxmg Jun 07 '16 at 14:34
@Suppen and immibis: I agree, I worded my answer according to the question, not the title. – KeyboardWielder Jun 07 '16 at 17:58
Since the question doesn't mention which base the numbers are in: 3!*3 = 18, 'twenty' in base 12. No 3 required on the right side, and it's the values three and twenty. – DeLacey Jun 07 '16 at 18:41
1

Technically speaking, you should show all your bases when doing cross base calculations. So it would be written $$3_{10} + 3_{10} = 20_3$$ which uses far too many 3s as well as two ones and two zeroes. – tfitzger Jun 07 '16 at 21:11
Every base is base 10 – Nick T Jun 07 '16 at 22:53
@Suppen It doesn't say about twenty; it says about 20. – EKons Jun 08 '16 at 17:11

score 35 · Answer 6 · edited Jun 07 '16 at 14:34

How would you get 20, using two threes?

Any out of the box, non-mathematical solutions are accepted.

How about an “in the box” solution?
You'd be lucky to not get 20, following the explicit formula published in U.S.[A.] Code, Title 18:

§471. Obligations or securities of United States. &hairsp; Whoever, with intent to defraud, falsely makes, forges, counterfeits, or alters any obligation or other security of the United States, shall be fined under this title or imprisoned not more than 20 years, or both.

Naturally, the threes are ...

... USA three-dollar bills ...

This formula can be generalized for any positive number of threes, of course, but may not translate exactly to all international units of measure.

Ooops, to be really sure to get 20, you'd have to first reassemble pieces of the threes to more closely resemble a denomination actually in circulation — humn, Jun 08 '16 at 03:24

score 27 · Answer 7 · answered Jun 06 '16 at 19:53

27

Here's another one:

Sorry for MS Paint skills

answered Jun 06 '16 at 19:53

Business Cat

1,914
13
23

I was thinking of using a couple of threes drawn with four matchsticks each (e.g. two stacked ">" each) but turning the first left 90 degrees and the second right 90 degrees, and then stacking them to yield "XX", but leaving the threes "intact". – supercat Jun 06 '16 at 20:12
Opening this spoiler crashes my iPhone app :O – Marco Bonelli Jun 07 '16 at 06:55
1

@MarcoBonelli: That's very strange, it's just a png image. – Business Cat Jun 07 '16 at 12:41
You can also arrange your halves to form $\rm XX$ if you allow curved $\rm X$'s. – yo' Jun 07 '16 at 22:06

James · Answer 8 · 2016-06-07T04:50:35.857

22

If we're allowing chopping up the numbers, how about some Roman Numerals

edited Jun 07 '16 at 04:50

answered Jun 07 '16 at 04:43

James

321
1
3

score 21 · Answer 9 · edited Mar 11 '17 at 02:47

21

As you specifically said to think outside the box, I drew a box, and two three's (one Roman numeral and one Arabic numeral) which extend "outside the box". As my answer note that they divide the box into 20 (yellow regions).

diagram

Therefore: Box / two three's = 20

edited Mar 11 '17 at 02:47

boboquack

21,982
1
66
138

answered Jun 08 '16 at 03:17

Stad

211
1
2

1

Isn't this an "inside the box" solution :) – Jun 08 '16 at 13:25
I <3 this one... haha, made my day. – tfrascaroli Jun 08 '16 at 13:46

score 19 · Answer 10 · answered Jun 07 '16 at 01:29

19

Two threes:

How would you make 20, using two threes?

Solution:

With apologies.

answered Jun 07 '16 at 01:29

user1717828

1,535
10
17

score 14 · Answer 11 · answered Jun 07 '16 at 08:56

14

How would you make 20, using two threes?

Here's the two: 2

and here are the threes: 3 - 3 = 0

giving 2 0.

answered Jun 07 '16 at 08:56

paolo

881
5
7

score 13 · Answer 12 · answered Jun 07 '16 at 01:34

13

A little bit silly, but it gets 20 without using any digits other than the two '3's

$\lfloor3+3+\pi+\pi+\pi+e+e\rfloor$

answered Jun 07 '16 at 01:34

BenM

1,273
8
11

Let me understand, are you claiming that irrational numbers are rational? ah, got it XD – CoffeDeveloper Jun 07 '16 at 09:12
6

@DarioOO He's just using the floor function (wolfram). – Marco Bonelli Jun 07 '16 at 12:55
@MarcoBonelli He meant $\pi$ and $e$. – EKons Jun 07 '16 at 16:25

score 11 · Answer 13 · answered Jun 08 '16 at 00:03

11

Since the OP says "any operation is allowed", I find it convenient to use this one: 3 ʭ 3, where the operator ʭ is defined as follows: ∀(x∈Z) ∀(y∈Z) x ʭ y = 20

answered Jun 08 '16 at 00:03

MattClarke

2,033
2
12
10

5

Ah, the old GiveTwenty operator. – Phil H Jun 08 '16 at 13:43

score 10 · Answer 14 · edited Jun 07 '16 at 14:36

10

output(++(++(++3))*(++3)); Consider this as a C expression

edited Jun 07 '16 at 14:36

JMP

35,612
7
78
151

answered Jun 07 '16 at 06:54

Mrigank Saini

109
2

6

error: lvalue required as increment operand – user253751 Jun 07 '16 at 07:10
for safety, use pre-increment – JMP Jun 07 '16 at 07:21
2

For correctness, use pre-increment. As written, the result, if the code were valid at all, would be 9. – Christopher Creutzig Jun 07 '16 at 08:37
I tried it with pre-increments and it still gives me an error, at least with clang: 3++.c:4:21: error: expression is not assignable – Jun 07 '16 at 14:27
You forgot the semicolon. – user1717828 Jun 07 '16 at 14:34
this needs variables use x=3,y=3 – JMP Jun 07 '16 at 14:39
Much shorter than my --(--(--(--(--(--(--(--(--(--(--(--(--33)))))))))))), good job – humn Jun 07 '16 at 19:46
1

On top of the compilation errors that others mentioned, this would produce 24, not 20. – Kevin Jun 08 '16 at 12:28
@Kevin right becaus of ++x instead of x++ thought it woul be 21 – Raoul Mensink Jun 08 '16 at 13:45

score 7 · Answer 15 · answered Jun 06 '16 at 19:53

7

I would sell the first $3$ to person A for $10$, and the second to Person B, again for $10$, giving me $20$

answered Jun 06 '16 at 19:53

JMP

35,612
7
78
151

7

Wouldn't you then have 14 since you had to give 6 away? :p – Inazuma Jun 07 '16 at 00:39
1

you would have 20 and sold 6. you could sell at 13 i suppose, but you would then have 26. depends how you look at it. – JMP Jun 07 '16 at 03:42
2

i had 6, i made 14 profit, now i have 20 – JMP Jun 08 '16 at 06:10

score 5 · Answer 16 · edited Jun 07 '16 at 14:40

5

Just buy two \$3 things from ebay. That will be \$20 easy.

Not answering within the spirit of this, sorry, but I'm waiting with baited breath as I can't figure out a correct answer.

edited Jun 07 '16 at 14:40

JMP

35,612
7
78
151

answered Jun 07 '16 at 14:08

LX07

59
1

1

You can display the $ sign by writing \$ – Red-Jeweled Jun 07 '16 at 14:15
2

baited breath -- ರ_ರ – user1717828 Jun 07 '16 at 18:38
@user1717828 I second that ರ_ರ – Kevin Jun 08 '16 at 12:30

score 5 · Answer 17 · answered Jun 07 '16 at 19:59

5

Very non-mathematical and taking a leaf out of the Car Talk match-stick puzzlers, we can shift around some lines and get a 2 and a 0 from a 3 and a 3.

answered Jun 07 '16 at 19:59

Jonas Powell

51
1

I would call that lateral thinking, for sure! – Engineer Toast Jun 07 '16 at 21:17

score 5 · Answer 18 · answered Jun 07 '16 at 20:47

5

Step 1: locate a person with a 20 dollar bill

Step 2: offer $33 for the 20 dollar bill

Step 3: acquire the 20 dollar bill in exchange for $33

Step 4: profit?? (probably not, unless it was some kind of vintage 20)

Congratulations you have converted \$33 into \$20!

answered Jun 07 '16 at 20:47

YoungJohn

901
5
9

4

"Step 4: profit???" just looks weird when we know that successful business models have "step 2: ???; step 3: profit" – Hagen von Eitzen Jun 08 '16 at 12:45

score 5 · Answer 19 · answered Jun 07 '16 at 20:58

5

In C++:

#include <cassert>
int ConvertThreesToTwenty(const int first_three, const int second_three)
{
    // verify preconditions
    assert(first_three == second_three && first_three == 3);
    return 20;
}

answered Jun 07 '16 at 20:58

YoungJohn

901
5
9

score 4 · Answer 20 · answered Jun 07 '16 at 15:08

4

If we consider the function :$$f(x) = (x-1)\sum_{i=1}^{x+1}i$$ Then we see that $f(3) = 20$. So in this case we get away with using just one 3, but maybe $-1$ and $+1$ inside the function would be considered cheating. If we must use two threes then function: $$g(x,y) = (x-1)\sum_{i=1}^{y+1}i$$ then $$g(3,3) = 20$$

Aren't the functions simple enough to be considered elementary operations?

answered Jun 07 '16 at 15:08

mathreadler

157
7

2

Then f(x,y) = x+y+14 is also simple enough. :) – KeyboardWielder Jun 07 '16 at 18:12
Yes a first degree polynomial is simpler than my second/third degree. But 14 is not as "basic" as +1 or -1. :) – mathreadler Jun 07 '16 at 18:32
1

^vote for using both variables where one could simply have been ignored. Could further substitute $\frac x y$ for each $1$ . . . – humn Jun 07 '16 at 19:30

score 4 · Answer 21 · edited Apr 13 '17 at 12:50

4

Using buttons found on most any calculator, hit the keys:

$3$ $!$ $!$ $\div$ $3$ $!$ $^2$ $=$

The result will be $20$.

I know that's a stretch on the definition of basic operations, but it works.

More in keeping with the rules, the following also works (from comment by humm):

$3$ $!$ $!$ $\div$ $3$ $!$ $=$ $=$

edited Apr 13 '17 at 12:50

Community

1

answered Jun 07 '16 at 17:54

user3294068

7,498
23
32

1

OP specifically stated that you can't directly raise to a digit other than 3. But perhaps you could abuse the features which repeat the previous operation/operand or use the M+/MR keys. ;) – KeyboardWielder Jun 07 '16 at 18:16
2

Your workaround works great, @KeyboardWielder, and compelled me ^vote this answer after trying: 3 n! n! / 3 n! = = (could be edited into the answer) – humn Jun 07 '16 at 19:19
You can always write just floor of e instead of 2. I had to do that in my solution. – BoltKey Jun 07 '16 at 21:50
I agree with @BoltKey. Floor e. – Athena Jun 09 '16 at 03:25
Just a note on notation. $n!!\ne (n!)!$ - the former is called semifactorial and is a notation for product of all positive integers up to $n$ which have the same parity as $n$. – elias Feb 26 '17 at 22:53

3:3! It's a football score!

21 Answers21