Blog Post #150: An Epicentre of Culture ♥

Dear All,

I had the pleasure this past week of meeting Maximilian Gililov, the youngest son of world-famous classical pianist Pavel Lvovich Gililov. Max was visiting Musica Mundi School, which he described beautifully as “an epicentre of culture”.

The following feast of fun puzzles is specially dedicated to Max 😊

1. Of the three chess games that we played, the third game lasted for 38 moves, the second for 39 moves, and the first game was shorter. The average number of moves in the three games was a whole multiple of ten.

Part 1: Exactly how many moves were played in the first game?

2. Rearrange the letters of US SECLUDED BAY or DEAD BUSY CLUES to get the name of Max’s favourite famous musical composer!

3. Max is an excellent pianist, but his other favourite instrument can be found by changing just one letter of HELLO to a different letter…

4. Choose well and replace one letter that is repeated in AIR STAR by the letter U, and then rearrange to make the seven-letter name of the beautiful country where Max lives and works.

5. Max and I thoroughly enjoyed the concerts that we saw. All the performances were superb! For every student who is practising to become better and better, Max’s honest words regarding himself can be inspirational. He said, “I’m an inferior version of my future self.”

I saw two concerts: one on Thursday in which every performance was on the piano; and one on Wednesday in which each performance was on either the harp, the oboe, or the cello.

For Thursday’s concert, the five-letter word PIANO was written 11 times (alongside 11 performers), making a total of 55 ‘piano’ letters there. For Wednesday’s concert, the total number of ‘HARP’, ‘OBOE’ & ‘CELLO’ letters was also 55, and there were more cellists than harpists or oboe-players.

Exactly how many cello performances were there in Wednesday’s concert? Also, what was the total number of performances (featuring harp, oboe, or cello) in that concert?

6. Max joked, “Life is not always black and white, but if you’re a pianist who likes to play chess, then it kind of is!!” 😊

7. Are you feeling nicely relaxed here in Blog Post #150?

Part 1: In our normal base ten, the number 150 ends with a zero. In which different number base does it end with two zeros, and exactly what other digits would be needed?

Part 2: The number 150 (base ten) consists of B digits when represented in base B. What is the value of B, and exactly what digits would be needed this time?

8. Imagine that Max and his parents have a water bottle each. All three bottles have the same capacity in litres: either 0.5, 0.75, 1, 1.5 or 2 litres.

Max’s bottle is full, but his parents’ bottles are empty. In what follows, the numbers of millilitres of water in the bottles are always whole numbers after each sharing action. (You can assume that everything is done very precisely with the aid of measuring cylinders and such items, if you like!) 😊

Max first pours some water from his own bottle to his father’s bottle until they both have equal amounts.

Max’s father then pours some water from his own bottle to his wife’s bottle until they both have equal amounts.

Max also pours some millilitres of water from his own bottle into his mother’s bottle, and then he gives his father the same nice bonus 😊.

All three bottles now contain exactly equal amounts of water.

Exactly how many millilitres of water did Max pour personally into his mother’s bottle? Also, what is the full capacity of each of the bottles?

9. Max was born in the year 2000. If you multiply the age he’ll be on his birthday this year by the number of days in his birth month, the three-digit result contains Max’s three favourite (positive one-digit) numbers 😊😊😊.

What are Max’s three favourite numbers?

10. The largest of Max’s three favourite numbers is his absolute favourite, and it’s also the month number for his birthday 😊. Now in 2023, day #1 was 1 January, day #42 is 11 February, and so forth. Max’s birthday in 2023 will be on day #N of this year, and N is a whole multiple of Max’s current age.

The fun challenge is to figure out Max’s exact date of birth.

11. List Max’s three favourite positive numbers from smallest to largest. Think of them as being the first three terms of a never-ending sequence of numbers! In that sequence, each term (from the second term onwards) can be generated by Multiplying the term immediately before it by M and then Adding on A.

Exactly what numbers do M and A represent?

Also, what is the exact value of the term in the sequence that is closest to 1000000?

12. Max went to high school in the lovely Austrian village of St. Gilgen, where Mozart’s mother was born loooooooooooooooooooong ago! 😊

It’s time now for one of my favourite brainteasers of all time, which I solved really happily in my early teens at school in Scotland 😊♥😊.

As Max will be turning 23 later this year, I’m thinking of a very special 23-digit whole number which starts and finishes with 1 at both ends. If you divide the super-loooooooooooooooooooong number by 99, the result is a twenty-one digit whole number which looks absolutely identical to the starting number except that it’s missing the first and last ‘end 1s’.

Your super-fun brainteaser is to find the 23-digit number in honour of Max! 😊

Notes:- No computer program or calculator is needed at all. The brainteaser can be solved quickly on paper. I find it a delightful bonus detail that there are 21 and 23 letters respectively in the names of Wolfgang Amadeus Mozart and his mother, Anna Maria Walburga Mozart 😊♥😊.

13. Last week, I gave a ruler as a gift to a Maths student. Now, today, I’m thinking of one of the greatest mathematicians of all time (with initials LE) whose full name can be found by rearranging the letters of LE HAD ONE RULER. The world-famous mathematical genius was born in the year – – – – and the digits are the same as Max’s birthday when stated in the form ddmm (day number followed by month number) 😊

Can you identify the famous mathematician?

14.

Imagine tiny me sitting on flat ground at the same level as the base of the Burj Khalifa, but some distance from it. Superman has flown to a point at a certain height on the skyscraper, and the eye-to-eye distance between him and me is 1000 feet. Max has already climbed to a point on the skyscraper that is 1000 feet directly above where Superman is.

When I look up at Max, his angle of elevation is twice (or double) the angle of elevation involved when I look at Superman.

Your sky-high brainteaser is to figure out the number of feet that Max still has to climb to reach the very top. (Just assume that people’s heights are negligible in comparison to the other distances involved in the puzzle.)

Fun Note: In a way, the digits of my answer are a tribute to Max and to Jens Van Steerteghem, an excellent mathematical/scientific colleague of mine at Musica Mundi School, as will be explained when the solutions are published.

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 ♥

With kindest wishes as always,

Paul M😊twani ♥

B😊NUS CHESS PUZZLE 😊

“And the spirit of the Lord shall rest upon him, the spirit of wisdom and understanding, the spirit of counsel and might, the spirit of knowledge and of the fear of the Lord”–Bible Verse, Isaiah 11:2 ♥

P.S. = Puzzle Solutions (being posted on 17.2.2023)

1. Part 1: 13 moves; Part 2: The possibilities are infinite! For example, draw a straight line from a point on the baseline of the photo, at a distance d from the bottom left-hand corner, to a point along the top of the photo at a distance d from the top right-hand corner. The rectangular photo is thereby split into two trapeziums of equal areas; the area of either one is exactly half of the area of the entire rectangular photo.
2. US SECLUDED BAY or DEAD BUSY CLUES →CLAUDE DEBUSSY !
3. HELLO – H + C →CELLO.
4. AIR STAR – R + U →AUSTRIA.
5. There were 7 cello performances and 5 harp or oboe performances, making 12 performances in total.
6. Hiding behind Max’s glass were the white queen and another white pawn. They were no longer on the board itself.
7. Part 1: 150 (base ten) is equivalent to 1100 (base five) or, for Part 2, 2112 (base four); B = 4.
8. Max personally poured 125 ml from his 1.5-litre bottle to his mother’s 1.5-litre bottle. Among the bottle-sizes mentioned in the puzzle, only the 1.5-litre option was wholly divisible by four and by three, so that Max’s parents could have a quarter-full bottle each (with 375 ml), before Max topped them up to one-third full (with 500 ml).
9. 23 x 31 = 713; Max’s three different favourite numbers are 7, 3 and 1.
10. Max’s exact date of birth was 17.7.2000. Note that, in 2023, July 17 will be day #198 of this non-leap year, and 198 = 9 x Max’s current age of 22.
11. The numbers in the sequence 1, 3, 7, … can be generated by multiplying terms by 2 and adding on 1 to get the next term; M = 2 & A = 1. Note also that each term is always 1 less than some power of 2. For instance, 3 = 2 squared – 1 & 7 = 2 cubed – 1. The value of the term that is closest to 1000000 is 1048575, which is 1 less than 2 raised to the power of 20.
12. The magical number from my early teens is 11123595505617977528091. Note that 11123595505617977528091 ÷ 99 = 112359550561797752809.
13. LE HAD ONE RULER →LEONHARD EULER, born in 1707.
14. In the puzzle, Superman is 500 feet up from the base of the Burj Khalifa at a 30° angle of elevation from where I am, while Max is 1500 feet up at a 60° angle of elevation with respect to me. Max still has a further 1217 feet to climb to reach the top, and the digits 12 & 17 are early celebrations of the birthdays of Jens Van Steerteghem & Max Gililov on July 12 and 17 respectively 😊😊
15. In the Chess puzzle, White won beautifully with 1 Bxf5 exf5 2 Nh6+ Kh8 3 Rxg7! Kxg7 4 Rg1+ Kh8 5 Qe2!! Be6 (or 5…Qxe2 6 Nxf7#) 6 Nxf7+!, intending 6…Qxf7 7 Qe5+.