Blog Post #159: Once and Forever ♥

Dear All,

I try to be grateful always for every moment that God grants, and this summer was filled with wonderful people, places, events and countless other blessings.

Just one of the special companions for me during three open international Chess tournaments–one in Germany and two in Belgium–was The Saints’ Little Book of Wisdom (compiled by Andrea Kirk Assaf).

Creative Puzzles Time!! ✔👌😎

Puzzle #1, Part 1

Puzzle #1, Part 2

Suppose that, in the diagram above, the dimensions r, h and L are all whole numbers of centimetres.

Part 2.1: Explain how you can then be absolutely sure that r, h and L must all be different, distinct numbers.

Part 2.2: Also, what is then the absolute minimum possible value for

the sum r + h + L?

Part 2.3 Brainteaser:

Thinking ahead with hope to blog post #160 (God-willing as always), can you discover suitable whole number values for r, h and L such that r + h + L = 160?

Puzzle #2

Think back to Blog Post #158…Can you discover a fitting reason for why June 7 was the perfect date for that post to be published?

Puzzle #3 (Brainteaser)

Suppose that Beethoven is writing down the same one-digit whole numbers that friends Jens, David, Jan, Herman, Peter and Paul are thinking of. Among us six seated friends, it turns out that we are all thinking of the same number…except for Paul! So, five of our numbers are identical; only Paul’s number is different. When Beethoven multiplies the six numbers together correctly, he gets a particular year that was just the second year of a certain century in the past. Exactly what year does Beethoven get in this puzzle!? 👏

Puzzle #4 (Sunset Girl Maths/Chess Mega-Brainteaser) 😍

Suppose that my niece in England and her dad play almost 100 Chess games together. To encourage really good, combative play, I award three points for a win, 1 point each for a draw, no points for a loss. After all the games have been played and points have been duly awarded according to the results, we calculate

Total Points of the Two Players ÷ Number of Chess Games they Played.

In their case, that turns out to be a decimal number a.b, in which ‘a’ is the whole number part, and ‘b’ is the one-digit decimal part.

If we convert the decimal number a.b to a fraction in its simplest form, we get c/d, where c and d are prime numbers, and a, b, c and d are all different.

Given the information above, your Sunset Girl Mega-Brainteaser in honour of my niece is to figure out the exact maximum-possible number of games of Chess that she played with her dad. Also, exactly how many of those games ended with a decisive win/loss result, and exactly how many of the games ended as draws?

Puzzle #5: Checkmate! 😃

It’s my intention to publish solutions to all the puzzles around the time that blog post #160 comes out, God-willing as always. (Before then, I will finally publish solutions to the puzzles of blog post #158 !)

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 by most sincerely wishing you a very blessed month of September soon, with lots of happiness in everything that you do ❤.

With kindest wishes as always,

Paul M😊twani ❤

“As gifts increase in you, let your humility grow, for you must consider that everything is given to you on loan, from God”–St. Pio of Pietrelcina.

“The secret of happiness is to live moment by moment and to thank God for all that He, in His goodness, sends to us day after day”–St. Gianna Beretta Molla.

P.S. = Puzzle Solutions (being published now on 30 October 2023)

CONE→ONCE.

Pythagoras’ Theorem tells us that L²=r²+h².

There are, of course, cases in which it’s possible for r and h to equal each other,

but then L² = 2r²

and so L = √2*r.

That’s not possible if L, r and h have to all be whole numbers (because √2, the square root of 2, is an irrational number).

Therefore, if L, r and h are positive whole-number dimensions on the cone, they must all be different from each other, and the slant height L must be greater than r and also greater than h.

In that case, the minimum-possible value for r+h+L is 3+4+5 = 12.

The consecutive whole numbers 3, 4, 5 form the most famous Pythagorean Triple: 3²+4²=5².

Whole numbers fitting the challenge r+h+L=160 are 32, 60 & 68. Note that 32²+60²=68². Also, the numbers 32, 60, 68 are four times larger than the famous 8, 15, 17 Pythagorean Triple in which 8+15+17=40.

June 7 was the perfect day for Blog Post #158 to be published because it was the 158^th day of 2023.

Beethoven would get 1701 (the second year of the 18th century) by correctly calculating 3 x 3 x 3 x 3 x 3 x 7.

In the brainteaser about Chess games, a.b = 2.6 = 13/5 = c/d.

a=2, b=6, c=13, d=5 with c and d being prime, as was stated in the puzzle.

60% of all the games end decisively and 40% of all the games are draws, which is why a.b = 2.6 or 3 – 0.4. (If all the games had ended decisively, then a.b would have reached its maximum-possible value of 3.)

The maximum-possible number of games (less than 100) is n=95, of which 60%→57 games are decisive wins and 40%→38 games are draws. Note: We wouldn’t get whole numbers there if n wasn’t a multiple of 5.

In the puzzle about Chess moves from 1962, White can win with 1 Qe6! Qxe6 2 Rxh7#, a neat checkmate! 👍😊💖