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₯

**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

**by multiplying their two numbers together.**

*product*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!** πβ₯π

**Additional Notes**:

1. The purposeful meaning behind the words ** ‘as well’ **above is that the sum and the product are

**in the 2023 whole numbers, with the sum at the very start and the product at the very end.**

*both included***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 **π

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

** 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, Γ· as you wish.** π

**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₯ππβ₯

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

**Puzzle #6: OUR ANGLE Brainteaser **π

**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 π.

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 Mπtwani 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 ππ

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 Γ· 2 = **2145**, which is nice here in blog post #145 π

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) ** πβ₯π

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! πβ₯π**

OUR ANGLE Brainteaser

*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 **πβ₯π**