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! ππ
After the concert, lots of people enjoyed mingling together and chatting at a very nice reception π
the parents of my brilliant colleague Jens Van Steerteghem ππππ
Word Puzzle
Hidden in the word ‘LATENT’ is another proper six-letter English word which all MMS students have… Have you found it already!? π
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 πΆπ.
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 π
We enjoyed playing our ‘royal game’, as well as discussing it afterwards
and finding multiple ways to improve the quality of our play ππ
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 π
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 ππ!!”
OR
you might say, “Can I have a drink instead?!” ππ
If you chose the latter, then…
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! π
BRAINTEASER π
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 π
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 π.
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 β€.
With kindest wishes as always,
Paul Mπtwani β€
The angel said to Mary, βThe 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 Liavβs favourite positive number, L. That is, a + b = L Λ 0. Weβll now use the very handy mathematical βidentityβ a3 + b3 = (a+b)3 β 3ab(a+b), which becomes a3 + b3 = L3 β 3PL, where P = the product ab. Itβs 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 β 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=1βL2=4βL=2 (not -2, since we require L Λ 0). Liavβs favourite number is indeed 2 ππ
Paul Motwani 