Blog Post #145: Happy New Year Brainteasers! πŸ˜Šβ™₯😊

Dear Readers,

My family and I hope that this message finds you keeping well, and we would like to wish you a very happy new year, 2023 β™₯

We’ve got some super-fun puzzles for you to enjoy solving! 😊β™₯😊

Puzzle #1: A Magical New Year Brainteaser!

Michael and Jenny write down two positive whole numbers. They calculate the sum by adding their two numbers together, and they also calculate the product by multiplying their two numbers together.

Guess how many whole numbers there are from their sum up to and including their product as well…there’s exactly 2023 whole numbers !

Your fun challenge is to discover Michael & Jenny’s numbers!

There are three different possible solutions! 😊β™₯😊

Additional Notes:

1. The purposeful meaning behind the words ‘as well’ above is that the sum and the product are both included in the 2023 whole numbers, with the sum at the very start and the product at the very end. Be extra-careful when considering the difference between the product and the sum…

2. Very sincere thanks to Teun Spaans (on 6 January 2023) for having kindly mentioned this brainteaser on his site justpuzzles.wordpress.com, but please note well point 1, just above.

Some Maths students know that 5! means 1 x 2 x 3 x 4 x 5 = 120.

Here in blog post #145, it’s nice to note that the special number 145 = 1!+4!+5!

Puzzle #2: A Good Book Puzzle β™₯

A boy is enjoying currently reading two of the 66 distinct books of The Bible that he has been given as a gift β™₯

Your puzzle is this: If you wanted to choose two out of 66 distinct books to read, how many different selections would be possible?

Puzzle #3: A Champion’s Challenge 😊

I recently received a happy message from Cansu, an excellent former Maths student of mine whom I always think of as ‘Champion Cansu’! 😊

In honour of C CANSU, use the numbers 3, 3, 1, 14, 19, 21 (in any order that you want) to make the target number 2023. You may also use parentheses ( ) and any of the operations +, -, x, ÷ as you wish. 😊

Puzzle #4: A Neat Word Puzzle

Rearrange the letters of the word LISTEN to make another proper six-letter English word.

There are four different possible solutions! β™₯😊😊β™₯

Puzzle #5: Dedicated to Elton (a former student of mine from 2012-2013 who likes the Royal Game of Chess) 😊

Black appears to be down on material by more than 2 knights…
but there’s an invisible black knight somewhere on the f-file
Can you discover two possible locations for it, so that in either case
it will then be Black to move and win by force!?

Puzzle #6: OUR ANGLE Brainteaser 😊

The OUR ANGLE picture is composed of several straight lines (ORGE, URA, LGA, NR, NG).
x and y are positive whole numbers. The angles xΒ°, 2xΒ°, yΒ°, 2yΒ° are not drawn to scale.

Your brainteaser is to figure out the maximum possible size (in degrees) for angle RAG that fits correctly with the given information.

It’s my intention to publish solutions to all the puzzles around the time that blog post #146 comes out, God-willing as always.

In the meantime, please do feel free to send me your best solutions to any or all of the puzzles, if you like 😊.

I would like to round off this article now by most sincerely wishing you a very blessed 2023, with lots of happiness in everything that you do β™₯

With kindest wishes as always,

Paul M😊twani β™₯

“You have decided the length of our lives. You know how many months we will live, and we are not given a minute longer. You set the boundary, and no one can cross it.”–Job 14:5

Joke: Why should you keep your left foot still at the beginning of January?

You’ll start off the New Year on the right foot!

P.S. = Puzzle Solutions (being posted now on 6 January 2023)

Magical New Year Brainteaser

If we let Michael & Jenny’s positive whole numbers be x and y, then their sum is x+y and their product is xy, which are respectively the very first and very last of 2023 consecutive whole numbers. Therefore, the difference xy – (x+y) = 2022 (not 2023). However, the puzzle can be solved really neatly by noting that xy – x – y + 1 = 2023, because we can then use factorisation to get that (x-1)(y-1) = 2023. In other words, 2023 is the product of whole numbers (x-1) and (y-1) which must both be factors of 2023 😊😊

2023 = 1 x 2023 or 7 x 289 or 17 x 119 (because 2023 = 7 x 17 x 17), and so we get that x-1 = 1 & y-1 = 2023 or x-1 = 7 & y-1 = 289 or x-1 = 17 & y-1 = 119, leading to

(x, y) = (2, 2024) or (8, 290) or (18, 120). Those are our 3 distinct solutions β™₯

Naturally, x & y are interchangeable, but in the context of this number puzzle we wouldn’t count 2024 & 2 (for example) as being a different pair from 2 & 2024.

A Good Book Puzzle

The number of different possible selections of two books from 66 distinct books is 66 x 65 ÷ 2 = 2145, which is nice here in blog post #145 😊

A Champion’s Challenge

We saw in the first puzzle that 2023 = 119 x 17, and having that target in mind can help us to use 3, 3, 1, 14, 19, 21 to easily make 2023 as follows:

((19 + 21) x 3 – 1) x (3 + 14) 😊β™₯😊

A Neat Word Puzzle

Listen = Silent = Tinsel = Enlist = Inlets β™₯

In the chess puzzle, if Black’s invisible knight is on f4, then 1…Ne2# delivers checkmate instantly!

Alternatively, if the invisible knight is on f2, then Black wins beautifully with 1…Rh1+! 2 Bxh1 Nh3+ 3 Kh2 Qe2+! and then either 4 Bg2 Qxg2# or 4 Kxh3 Qh5#, a very pretty checkmate! 😊β™₯😊

OUR ANGLE Brainteaser

The OUR ANGLE picture is composed of several straight lines (ORGE, URA, LGA, NR, NG).
x and y are positive whole numbers. The angles xΒ°, 2xΒ°, yΒ°, 2yΒ° are not drawn to scale.

Note: In all of the following steps of working, the angle sizes are in degrees.

Step 1: Angle NRG = 180 – (2x + y) & angle NGR = 180 – (2y + x).

Step 2: Angle RNG = 180 – angle NRG – angle NGR; after now using results from Step 1 above, and simplifying the algebraic terms, we get that angle RNG = 3x + 3y – 180 or, in factorised form, angle RNG = 3(x + y – 60).

Step 3: From the result of Step 2 above, we can conclude that x + y must be greater than 60.

Step 4: Angle ARG = 2x & angle AGR = 2y, so angle RAG = 180 – 2x – 2y, which can also be written as 180 – 2(x + y).

Step 5: Since x + y is greater than 60 (from Step 3 above), angle RAG must be less than 180 – 2 (60); so angle RAG must be less than 60 degrees. That would normally be our final conclusion… However…

Step 6: Since we were given that x and y are positive whole numbers, then the minimum possible value for x + y is 61, to be greater than 60.

Therefore, when x & y are whole numbers,

the maximum possible measure for angle RAG is 180 – 2 (61) = 180 – 122 = 58Β°.

(It’s also worth noting that, when angle RAG = 58Β°, angle RNG = 3Β°.)

Very warm congratulations to everyone who enjoyed trying and solving some or all of the puzzles 😊β™₯😊

Author: Paul A. Motwani

My name is Paul Motwani, but my colleagues, my students and their parents mostly call me "Mr. Mo"! My middle initial, A, stands for Anthony, because I was born on the official feast day of St. Anthony of Padua, the patron saint of miracles and of lost souls. I love teaching Mathematics and Chess, and giving fun-packed talks and shows in schools and clubs. The popular ingredients of Math, Chess, Mystery and Magic are my "Fantastic Four", and I give prizes too! I am an International Chess Grandmaster, and (loooooong ago!) I was the World Under-17 Champion. I am the author of five published chess books and hundreds of newspaper articles. I live with my wonderful wife and son in Belgium. I also love music, movies and puzzles. I blog at paulmotwani.com. My e-mail address is pmotwani141@gmail.com. You can find me on Facebook, too.

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