Saturday, April 1, 2023
HomePuzzleA Binary Operation Puzzle – Thoughts Your Selections

# A Binary Operation Puzzle – Thoughts Your Selections Due to Boomii from Taiwan for the suggestion!

Suppose the binary operation ☆ is outlined in order that

xy = (x + y) / (1 + xy).

Simplify the expression

{ … [(2 ☆ 3) ☆ 4] ☆ … } ☆ 2021

As normal, watch the video for an answer.

Binary operation puzzle

Or preserve studying.
.
.

“All will probably be nicely should you use your thoughts in your choices, and thoughts solely your choices.” Since 2007, I’ve devoted my life to sharing the enjoyment of sport idea and arithmetic. MindYourDecisions now has over 1,000 free articles with no adverts because of group assist! Assist out and get early entry to posts with a pledge on Patreon.

.
.

.
.
.
.
M
I
N
D
.
Y
O
U
R
.
D
E
C
I
S
I
O
N
S
.
P
U
Z
Z
L
E
.
.
.
.
Reply To A Binary Operation Puzzle

(Just about all posts are transcribed shortly after I make the movies for them–please let me know if there are any typos/errors and I’ll appropriate them, thanks).

Let’s work out a number of examples.

2 ☆ 3 = (2 + 3)/(1 + 2·3) = 5/7
(2 ☆ 3) ☆ 4 = (5/7 + 4)/(1 + (5/7)·4) = 9/11
[(2 ☆ 3) ☆ 4] ☆ 5 = (9/11 + 5)/(1 + (9/11)·5) = 7/8 = 14/16
{[(2 ☆ 3) ☆ 4] ☆ 5} ☆ 6 = (7/8 + 6)/(1 + (7/8)·6) = 11/10 = 22/20

Let an be the expression with the most important integer n. Then from the calculations, we will hypothesize a method:

if n is odd, then
an = (n(n + 1)/2 – 1)/(n(n + 1)/2 + 1)
if n is even, then
an = (n(n + 1)/2 + 1)/(n(n + 1)/2 – 1)

Assuming the method is true, we’ve:

a2021
= (2021(2021 + 1)/2 – 1)/(2021(2021 + 1)/2 + 1)
= 1,021,615/1,021,616

Proof of method

Let’s use mathematical induction. The method is true for base instances n = 3, 4. Now assume the method is true for an integer okay &geq;3. We have to present the method holds for okay + 1. We are going to break the proof into instances the place okay is odd and even.

If okay is odd, then okay + 1 is even and we’ve:

aokay + 1 = aokay ☆ (okay + 1)

By the induction speculation we will substitute for aokay.

= [(k(k + 1)/2 – 1)/(k(k + 1)/2 + 1)] ☆ (okay + 1)
= [(k(k + 1) – 2)/(k(k + 1) + 2)] ☆ (okay + 1)
= [(k(k + 1) – 2)/(k(k + 1) + 2) + (k + 1)]/[1 + (k(k + 1) – 2)/(k(k + 1) + 2)(k + 1)]
= [(k(k + 1) – 2) + (k + 1)(k(k + 1) + 2)]/[(k(k + 1) + 2) + (k(k + 1) – 2)(k + 1)]
= (okay3 + 3okay2 + 4okay)/(okay3 + 3okay2)
= (okay2 + 3okay + 4)/(okay2 + 3okay)
= ((okay + 1)(okay + 2) + 2)/((okay + 1)(okay + 2) – 2)
= ((okay + 1)(okay + 2)/2 + 1)/((okay + 1)(okay + 2)/2 – 1)

This verifies the method holds for okay + 1.

The proof for okay is even is analogous (particulars omitted).

## Revealed by

### PRESH TALWALKAR

I run the MindYourDecisions channel on YouTube, which has over 1 million subscribers and 200 million views. I’m additionally the writer of The Pleasure of Recreation Idea: An Introduction to Strategic Pondering, and a number of other different books which can be found on Amazon.

(As you may count on, the hyperlinks for my books go to their listings on Amazon. As an Amazon Affiliate I earn from qualifying purchases. This doesn’t have an effect on the value you pay.)

By the use of historical past, I began the Thoughts Your Selections weblog again in 2007 to share a little bit of math, private finance, private ideas, and sport idea. It has been fairly a journey! I thank everybody that has shared my work, and I’m very grateful for protection within the press, together with the Shorty Awards, The Telegraph, Freakonomics, and lots of different in style retailers.

I studied Economics and Arithmetic at Stanford College.

Folks typically ask how I make the movies. Like many YouTubers I exploit in style software program to organize my movies. You possibly can seek for animation software program tutorials on YouTube to discover ways to make movies. Be prepared–animation is time consuming and software program may be costly!

Be at liberty to ship me an e mail [email protected]. I get so many emails that I’ll not reply, however I save all strategies for puzzles/video subjects.

### MY BOOKS

If you buy by means of these hyperlinks, I could also be compensated for purchases made on Amazon. As an Amazon Affiliate I earn from qualifying purchases. This doesn’t have an effect on the value you pay.

E book rankings are from January 2022.

https://mindyourdecisions.com/weblog/my-books Thoughts Your Selections is a compilation of 5 books: The Pleasure of Recreation Idea exhibits how you need to use math to out-think your competitors. (rated 4.2/5 stars on 224 critiques)

40 Paradoxes in Logic, Likelihood, and Recreation Idea incorporates thought-provoking and counter-intuitive outcomes. (rated 4.1/5 stars on 38 critiques)

The Irrationality Phantasm: How To Make Sensible Selections And Overcome Bias is a handbook that explains the various methods we’re biased about decision-making and presents strategies to make good choices. (rated 4/5 stars on 24 critiques)

The Finest Psychological Math Tips teaches how one can appear to be a math genius by fixing issues in your head (rated 4.2/5 stars on 76 critiques)

Multiply Numbers By Drawing Strains This ebook is a reference information for my video that has over 1 million views on a geometrical technique to multiply numbers. (rated 4.3/5 stars on 30 critiques)

Thoughts Your Puzzles is a group of the three “Math Puzzles” books, volumes 1, 2, and three. The puzzles subjects embrace the mathematical topics together with geometry, likelihood, logic, and sport idea.

Math Puzzles Quantity 1 options traditional mind teasers and riddles with full options for issues in counting, geometry, likelihood, and sport idea. Quantity 1 is rated 4.4/5 stars on 87 critiques.

Math Puzzles Quantity 2 is a sequel ebook with extra nice issues. (rated 4.1/5 stars on 24 critiques)

Math Puzzles Quantity 3 is the third within the collection. (rated 4.2/5 stars on 22 critiques)

### KINDLE UNLIMITED

Academics and college students world wide typically e mail me concerning the books. Since training can have such a big impact, I attempt to make the ebooks accessible as broadly as attainable at as low a value as attainable.

At the moment you’ll be able to learn most of my ebooks by means of Amazon’s “Kindle Limitless” program. Included within the subscription you’ll get entry to thousands and thousands of ebooks. You do not want a Kindle machine: you’ll be able to set up the Kindle app on any smartphone/pill/laptop/and many others. I’ve compiled hyperlinks to packages in some international locations beneath. Please verify your native Amazon web site for availability and program phrases.

### MERCHANDISE

Seize a mug, tshirt, and extra on the official web site for merchandise: Thoughts Your Selections at Teespring.

RELATED ARTICLES