Month: March 2023

Blog Post #153: A Super High Five For All MMS Musicians πŸ‘πŸŽΆπŸ’–

Dear All,

Just a few hours ago, hundreds of happy people were enthralled by all the marvellous student chamber music performances that they witnessed in the Bach Concert Hall within Musica Mundi School (MMS), Waterloo. Everyone who was involved deserves really warm congratulations for yet another stunningly beautiful production at the school! πŸ˜πŸ‘Œ

After the concert, lots of people enjoyed mingling together and chatting at a very nice reception 😊

It was a great pleasure for me to meet Martine & Eric,
the parents of my brilliant colleague Jens Van Steerteghem 😊😊😊😊

Steven, a fabulous 18-year-old Belgian pianist, is pictured above with his father 😊😊😊

Word Puzzle

Hidden in the word ‘LATENT’ is another proper six-letter English word which all MMS students have… Have you found it already!? 😊

I would like to take this opportunity to wish ThaΓ―s and Liav lots of enjoyment and success this coming Sunday when they’ll be performing together in Gent as part of a huge music festival πŸŽΆπŸ’•.

Super cellists ThaΓ―s & Liav πŸŽΆπŸ’•
ThaΓ―s is an outstanding alumna of Musica Mundi School,
and Liav, an excellent Chamber Music Faculty Member, is the son of the school’s founders,
Leonid Kerbel & Hagit Hassid-Kerbel.

The Chess set in the background of the above photo was well used this past Tuesday evening when TimothΓ©e and I played a ‘friendly’ game as a practice warm-up for a fun Chess event that we’re going to be having at the school later today, open to everyone in the MMS Family 😊

TimothΓ©e is one of the most talented student chess players at MMS πŸ‘
The Chess photo shows the final position from the game with TimothΓ©e (White) against me.
We enjoyed playing our ‘royal game’, as well as discussing it afterwards
and finding multiple ways to improve the quality of our play 😊😊

A ‘FEEL’ Number Puzzle

ThaΓ―s told me that her favourite number is a particular two-digit whole number which she likes for the ‘feel’ of it. Let’s represent that two-digit number by EL. It’s special because when it’s squared the result is the four-digit number FEEL, in which FE is exactly three-quarters of EL.

Your fun puzzle is to figure out the exact value of ThaΓ―s’s favourite number 😊

If I tell you that what’s coming next is an advanced algebra brainteaser at the request of a certain brilliant colleague, you’ll probably either react with something like, “Oh yes, that’s exactly what I wanted too πŸ˜πŸ˜‰!!”

OR

you might say, “Can I have a drink instead?!” πŸ˜œπŸ˜‚

If you chose the latter, then…

…a wee trip to Peter’s TeaHouse in Cremona, Italy, is highly recommended! πŸ‘πŸ˜

If you opted bravely for the brainteaser, then brace yourself…it’s coming now!

Here in Blog Post #153, we ought to note that 1 cubed + 5 cubed + 3 cubed = 153, and so it’s fitting that the brainteaser will feature cubes in it… Liav will be starring too, to make it nicer! πŸ˜ƒ

BRAINTEASER 😜

Liav knows that there’s an infinite choice of pairs of real numbers such that the sum of each pair equals his favourite number (which is positive). For each such pair, imagine calculating the sum of the cubes of the two numbers in the pair. It turns out that the absolute minimum possible sum of the two cubes equals Liav’s favourite number precisely!

Your brainteaser is to figure out, with proof, what Liav’s favourite number is 😎

It’s my intention to publish solutions to all the puzzles around the time that blog post #154 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 month of April coming soon, with lots of happiness in everything that you do ❀.

With kindest wishes as always,

Paul M😊twani ❀

The angel said to Mary, β€œThe Holy Spirit will come to you, and the power of the Most High God will cover you. The baby will be holy and will be called the Son of God.”

–Bible verse, Luke 1:35 β™₯

P.S. = Puzzle Solutions (being posted on 4.4.2023)

LATENT β†’ TALENT

Considering the place value of each digit in FEEL, the total value of FEEL = FE00 + EL = FE x 100 + EL, and (given information that FE = 3/4 of EL) that’s EL x 3/4 x 100 + EL, which simplifies to EL x 75 + EL β†’ EL x 76. So, EL squared = EL x 76, and therefore EL equals 76, ThaΓ―s’ favourite number.

Suppose that a and b are real numbers whose sum equals Liav’s favourite positive number, L. That is, a + b = L Λƒ 0. We’ll now use the very handy mathematical β€˜identity’ a3 + b3 = (a+b)3 – 3ab(a+b), which becomes a3 + b3 = L3 – 3PL, where P = the product ab. It’s a well-known result that (when numbers have a fixed sum) the maximum-possible value of their product P occurs when a = b, and here that would mean a = b = L/2, and Pmax = L/2 x L/2 = L2/4. So, the minimum-possible value of a3 + b3 is L3 – 3(L2/4)L β†’ L3/4. We were also given information implying that (a3 + b3)min = L, and therefore L3/4 = L β†’ L2/4=1β†’L2=4β†’L=2 (not -2, since we require L Λƒ 0). Liav’s favourite number is indeed 2 😊😊

Blog Post #152: A Good Heart πŸ’–

Dear All,

This article is specially dedicated to Dr. Vipin Zamvar, a consultant cardiothoracic surgeon who certainly has a really good heart, in the kindest, very best sense ❀. Vipin is originally from Mumbai, India, but now lives in Edinburgh, and my family and I had the pleasure of meeting the gentleman doctor in Scotland’s beautiful capital city through an event at Edinburgh Chess Club last October. We thoroughly enjoyed chatting with Vipin during dinner that evening, and when he kindly gave us a lift to our hotel afterwards. In addition to sharing the ‘Royal Game’ of Chess as a fine hobby, we also like Mathematics, and I am currently reading ‘The Music of the Primes’ which Vipin sent as a lovely gift book.

I would like to now offer several fresh brainteasers for the enjoyment of Vipin and all puzzle fans! 😊❀😊

  1. Start with the number 152 here in Blog Post #152.

Multiply it by my favourite number, 3, and then add 3.

If you divide the result by Vipin’s favourite whole number, you’ll then have a prime number.

What exactly is Vipin’s favourite whole number (given that it is not more than 152) ?

2. Rearrange the letters of CREMONA (a beautiful city in Italy) to make a proper 7-letter English word. The nice seven-letter word – – – – – – – has a connection to Vipin because he chose his favourite number for the reason that it was part of the date on which he first met his wife ❀

What a lovely couple! We see Vipin and his wife, Usha, pictured in Coimbatore, India πŸ’–πŸ’–

3. We already encountered the number 459 in the first puzzle (when doing 152 x 3 + 3), and now imagine that Vipin selects either 4 or 5 or 9. Let’s call his selected number V. Vipin will raise V to the power of his wife’s age now, and he’ll note the result. Vipin will also raise V to the power of his wife’s future age on her next birthday, and again he’ll note the result. Vipin will add his two results together to get a new, larger result, Z.

What exactly will be the units (or ‘ones’) digit of the number Z?

Can you prove what the digit will be?

4. Imagine a long bus travelling at a constant speed through a tunnel in India that is nearly 1km long. (The tunnel length is in fact a whole number of metres between 900 and 1000. The length of the bus is also a whole number of metres.) From the moment that the front of the bus enters the tunnel, the time taken for the entire bus to be inside the tunnel is t seconds. However, the time taken for the entire bus to pass through the tunnel is t minutes.

What is the exact length of the tunnel?

5. Now it’s time for an ABCD puzzle to wish you A Beautiful Creative Day!

😊❀❀😊

The diagram shows two overlapping circles of equal radii, r, say. The points A, B, C and D are collinear, and all lie on the line which passes through the centres of the circles.

If BC is not less than AB + CD,

then

what is the maximum-possible value for AD Γ· r ?

6. People don’t normally like going round in circles, but still…

…this next puzzle is actually lots of fun, too!! πŸ˜‚

Imagine that the distinct positive whole numbers 2, 3, 4, 5, 6 and X are going to be placed in the six rings; one number per ring. The products of the numbers on each of the three edges of the triangular array are to be equal to each other, and will each be P, say.

What is the value of X?

Also, what is the maximum-possible value for P?

7. In Chess, Vipin and I both like playing the Caro-Kann Defence as Black. So, let’s now enjoy seeing it in action in a super-fast victory 😎 from Kiev in 1965, the year when Vipin was born. πŸ’–

Mnatsakanian vs. Simagin, Kiev 1965.

1 e4 c6 2 Nc3 d5 3 d4 dxe4 4 Nxe4 Nf6 5 Nxf6+ exf6 6 Bc4 (6 c3 followed by Bd3 is more popular nowadays) 6…Be7 (Several decades ago, super-GM Julian Hodgson told me that he likes 6…Qe7+, especially if White responds with 7 Be3?? or 7 Ne2?? which lose to 7…Qb4+! 😁) 7 Qh5 0-0 8 Ne2 g6 9 Qh6 Bf5 10 Bb3 c5 11 Be3 Nc6 12 0-0-0? (White’s king castles into an unsafe region where it will be attacked very quickly indeed…) 12…c4!! 13 Bxc4 Nb4 14 Bb3 Rc8 (the point of Black’s energetic pawn-sacrifice at move 12 has become clear along the opened c-file) 15 Nc3 Qa5 (also good is 15…b5, intending 16 a3 Rxc3!! 17 bxc3 Nd5 with an enduring attack for Black in addition to having enormous positional compensation for the sacrificed material) 16 Kb1? (It’s often difficult to defend well against a sudden attack, but this move simply loses by force; 16 Bd2 is more tenacious)

Get ready for a beautiful Chess combination! 😍

16…Rxc3! (16…Bxc2+! 17 Bxc2 Rxc3 also works) 17 bxc3 Bxc2+! 0:1. White resigned, in view of 18 Bxc2 Qxa2+ 19 Kc1 Qxc2# or 18 Kc1 Nxa2+ with decisive threats against the fatally exposed White monarch.

I’m pretty sure that Scott Fleming (who recently sent me a really nice letter from Arbroath, Scotland) will also enjoy that very neat, crisp win for Black, as will FIDE Master Craig SM Thomson, who has played lots of wonderful games with the Caro-Kann Defence for nearly 50 years already!! πŸ‘πŸ˜Š

It’s my intention to publish solutions to all the puzzles around the time that blog post #153 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 weekend, with lots of happiness in everything that you do ❀.

Special congratulations to my friend James Pitts who has turned 53 today.

πŸŽ‚πŸ’–πŸ˜Š

With kindest wishes as always,

Paul M😊twani ❀

“He has dethroned rulers and has exalted humble people.”

–Bible verse, Luke 1:52 β™₯

P.S. = Puzzle Solutions (being posted on 4.4.2023)

  1. 459 Γ· 27 = 17, a prime number. Vipin’s favourite number is 27.
  2. CREMONA β†’ ROMANCE ! β™₯
  3. Note that VA + V(A+1) = VA(1+V). If V=4, then (1+v) = 5, and VA(1+V) will end with a digit zero. If V=5, then (1+v) = 6, and VA(1+V) will be even and will end with a digit zero (rather than a 5). If V=9, then (1+v) = 10, and again VA(1+V) will end with a digit zero.
  4. If B and T represent the respective lengths (in metres) of the long bus and the tunnel, then (since t minutes is 60 times longer than t seconds), we can deduce that B+T = 60 x B and so T = 59 x B, a whole multiple of 59. The only such value for T between 900 and 1000 is 59 x 16 = 944. The tunnel is 944 metres long (and the length of the long bus is 16 metres).
  5. If AD = L, then AB = AD – BD = L – 2r, and similarly CD = L -2r. Also, BC = L – 2(L-2r), which simplifies to 4r – L. So, for BC to be not less than AB + CD, we require that 4r – L β‰₯ 2 (L – 2r). That inequality can be simplified to 8r β‰₯ 3L, and so L/r ≀ 8/3. Therefore, if BC is not less than AB + CD, the maximum-possible value of AD Γ· r is 8/3.
  6. The only possible value for X is 10, and a maximum-possible product of 120 along each edge can be achieved by putting the numbers 4, 6 and 10 in the corner positions. Then, place 3 between 4 & 10; place 2 between 6 & 10; place 5 between 4 & 6. The products P each become 120.