Can You Solve The Three 3s Challenge? – Mind Your Decisions (2024)

Thanks to Yale from Hong Kong for suggesting this puzzle! The problem went viral after a video from Carl Ho (in Chinese): https://youtu.be/6acBMe0Ugy8.

Here is the problem: using three 3s, can you make the numbers from 0 to 10 using mathematical operations?

3 3 3 = 0
3 3 3 = 1
3 3 3 = 2
3 3 3 = 3
3 3 3 = 4
3 3 3 = 5
3 3 3 = 6
3 3 3 = 7
3 3 3 = 8
3 3 3 = 9
3 3 3 = 10

There are many ways to solve this, and I will present some of answers.

The hardest problem is 3 3 3 = 10. My challenge, which is a very hard challenge, is to solve this in two ways using only the following symbols:

+ ! ()

There is a way to solve 3 3 3 = 10 using only those symbols, although the notation is lesser known.

Can you figure it out? Watch the video for a solution.

Can You Solve The Three 3s Challenge?

Or keep reading.
.
.

"All will be well if you use your mind for your decisions, and mind only your decisions." Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon.

(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).

Here are some ways to get 0 to 9. (You may have found other solutions not listed here.)

3! – 3 – 3 = 0

(3! – 3)/3 = 1
3^{3 – 3} = 1
√3 × √3 / 3 = 1

3 – 3/3 = 2
(3 + 3)/3 = 2

3 + 3 – 3 = 3
3 × 3/3 = 3

3 + 3/3 = 4
3! – 3!/3 = 4

3! – 3/3 = 5
3!/3 + 3 = 5

3 × 3 – 3 = 6
3! × 3/3 = 6
3 << (3/3) = 3×2¹ = 6 (<< is the left arithmetic shift)

3! + 3/3 = 7
3! + (3 – 3)! = 7
3/.3 – 3 = 7

3!/3 + 3! = 8

3 + 3 + 3 = 9
3³/3 = 9

How can we make 10? Here are some of the creative ways I found.

In Carl Ho’s video comments, someone suggested:

arctan(√3)/(3 + 3) = 10 (degrees)

The idea is that arctan(sqrt(3)) is 60 degrees, so you then divide by 6 to get 10. The answer is technically in degrees, but for this puzzle we may allow this creative answer.

This is an answer from Cut The Knot:

floor(√3 + √3)×3 = 10

Yale suggested three methods to me as well.

√(√(3 × 3)/3%) = 10

log_√√3(3) + 3! = 10

3 × 3 + (d/dx (3))! = 10

Now for the challenge: how can we solve it using only +, ! and ( )?

Here are the solutions:

3! + !3 + !3 = 6 + 2 + 2 = 10

!3(!3 + 3) = 2(2 + 3) = 10

How is !3 = 2? This requires a bit of explanation.

The notation n! denotes the factorial of n, which is the total number of permutations of n.

By contrast, the notation !n denotes the subfactorial of n, which is the number of derangements of n. A derangement is a permutations in which no item is in its natural order.

For example,

!1 = 0 because {1} has no derangements

!2 = 1 because {1, 2} has the single derangement {2, 1}

!3 = 2 because {1, 2, 3} has two derangements: {2, 3, 1} and {3, 1, 2}

While this might be new to you, it is accepted enough notation that even WolframAlpha will evaluate. You can see:

!3(!3 + 3) evaluates to 10

It’s a sneaky way to use the exclamation point, but in this puzzle the subfactorial is very useful! You’ll also find extra solutions to the other equations using !3 = 2.

Another bonus answer

I received this by email:

!(!3 + !3) + !(!3) = 10

This uses that !4 = 9 and !2 = 1 which is really clever!

(Note: you can get the number of derangements by the formula floor(n!/e + 1/2) for n ≥ 1. So you will note the number of derangements, divided by the number of permutations n!, is roughly 1/e &approx; 37%. It’s kind of interesting the ratio of derangements to permutations is roughly constant even as n increases!)

Sources

Carl Ho’s video (Chinese)
https://youtu.be/6acBMe0Ugy8

Cut The Knot Representation of numbers with three 3’s
http://www.cut-the-knot.org/arithmetic/funny/3_3.shtml

Thanks to all patrons! Special thanks to:

Shrihari Puranik
Yildiz Kabaran
Kyle

Share the beauty of math with the world! You can support these videos and posts at Patreon and get exclusive rewards: http://www.patreon.com/mindyourdecisions

MY BOOKS

If you purchase through these links, I may be compensated for purchases made on Amazon. As an Amazon Associate I earn from qualifying purchases. This does not affect the price you pay.

Book ratings are from January 2023.

(US and worldwide links)
https://mindyourdecisions.com/blog/my-books

Mind Your Decisions is a compilation of 5 books:

(1) The Joy of Game Theory: An Introduction to Strategic Thinking
(2) 40 Paradoxes in Logic, Probability, and Game Theory
(3) The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias
(4) The Best Mental Math Tricks
(5) Multiply Numbers By Drawing Lines

The Joy of Game Theory shows how you can use math to out-think your competition. (rated 4.3/5 stars on 290 reviews)

40 Paradoxes in Logic, Probability, and Game Theory contains thought-provoking and counter-intuitive results. (rated 4.2/5 stars on 54 reviews)

The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias is a handbook that explains the many ways we are biased about decision-making and offers techniques to make smart decisions. (rated 4.1/5 stars on 33 reviews)

The Best Mental Math Tricks teaches how you can look like a math genius by solving problems in your head (rated 4.3/5 stars on 116 reviews)

Multiply Numbers By Drawing Lines This book is a reference guide for my video that has over 1 million views on a geometric method to multiply numbers. (rated 4.4/5 stars on 37 reviews)

Mind Your Puzzles is a collection of the three "Math Puzzles" books, volumes 1, 2, and 3. The puzzles topics include the mathematical subjects including geometry, probability, logic, and game theory.

Math Puzzles Volume 1 features classic brain teasers and riddles with complete solutions for problems in counting, geometry, probability, and game theory. Volume 1 is rated 4.4/5 stars on 112 reviews.

Math Puzzles Volume 2 is a sequel book with more great problems. (rated 4.2/5 stars on 33 reviews)

Math Puzzles Volume 3 is the third in the series. (rated 4.2/5 stars on 29 reviews)

KINDLE UNLIMITED

Teachers and students around the world often email me about the books. Since education can have such a huge impact, I try to make the ebooks available as widely as possible at as low a price as possible.

Currently you can read most of my ebooks through Amazon's "Kindle Unlimited" program. Included in the subscription you will get access to millions of ebooks. You don't need a Kindle device: you can install the Kindle app on any smartphone/tablet/computer/etc. I have compiled links to programs in some countries below. Please check your local Amazon website for availability and program terms.

US, list of my books (US)
UK, list of my books (UK)
Canada, book results (CA)
Germany, list of my books (DE)
France, list of my books (FR)
India, list of my books (IN)
Australia, book results (AU)
Italy, list of my books (IT)
Spain, list of my books (ES)
Japan, list of my books (JP)
Brazil, book results (BR)
Mexico, book results (MX)

MERCHANDISE

Grab a mug, tshirt, and more at the official site for merchandise: Mind Your Decisions at Teespring.