Blog Post #61: For Christopher & Kristina!

Dear Readers,

I very much hope that you will all enjoy this article, which is especially dedicated to Christopher (one of my first students at my former school in Belgium, nearly 15 years ago) and to Kristina, an outstanding student at the beautiful Music school where I now work as the Mathematics teacher.

Christopher’s family have made positive use of social media to publish good posts which help to raise people’s awareness of Mental Health matters. That is an important and complex subject which can potentially impact everyone. Here on this site, all the puzzles I post are a small personal contribution that I hope will bring enjoyment and healthy mental stimulation to all readers.

CHRISTOPHER’S BIRTHDAY BONUS!

Christopher already celebrated his birthday this month. You can quickly figure out when it was, if I tell you that Christopher’s day-number within November is equal to the square root of the number of days remaining in November after Christopher’s birthday.

Kristina is a world-class violinist who loves getting to play an almost magical instrument from the year 1737. A couple of days ago, the number 1737 starred in a fun Maths SMS that I sent to a colleague and to Kristina. SMS is a palindrome (which reads the same forwards as backwards), and 17362+17372+17382=9051509, a palindromic number.

Some High School Maths students would easily manage to prove that, though the sum of any three consecutive whole numbers is always a multiple of 3, the sum of the squares of any three consecutive whole numbers is never a multiple of 3. Just very occasionally, though, the sum of the squares does give a palindromic result, like 9051509.

Here are other such cases:-

42+52+62=77

112+122+132=434

372+382+392=4334

5662+5672+5682=964469

(17362+17372+17382=9051509, already given)

83392+83402+83412=208666802

160842+160852+160862=776181677

175522+175532+175542=924323429

179322+179332+179342=964777469, and I reckon that’s the last such case where the result is under 1 billion.

ONE BILLION TRICK CHALLENGE!

Your “one billion trick challenge” is to determine exactly how many magic triples (like Kristina’s 1736, 1737, 1738) there are for which the sum of the squares gives a palindromic result between one billion and two billion !

Solutions to the puzzles will be given at the time of the next blogpost.

As you’ll be considering “magic triples”, and since 3 is my absolute favourite number, my wife and my son and I would like to wish you all a very happy November now.

CHECKMATE BONUS!!

Consider the chess position shown below.

Imagine that White and Black each have one extra, invisible pawn somewhere on the board. You get to decide exactly where the two invisible pawns will be, but not anywhere on the 7th rank (e.g. NOT on d7 beside Black’s king). You also get to decide whose turn it will be to move. The challenge is to place the two invisible pawns so that one side can then force checkmate in as few moves as possible.

Wishing you lots of enjoyment,

Paul Motwani.

Author: Paul A. Motwani

My name is Paul Motwani, but my colleagues, my students and their parents mostly call me "Mr. Mo"! My middle initial, A, stands for Anthony, because I was born on the official feast day of St. Anthony of Padua, the patron saint of miracles and of lost souls. I love teaching Mathematics and Chess, and giving fun-packed talks and shows in schools and clubs. The popular ingredients of Math, Chess, Mystery and Magic are my "Fantastic Four", and I give prizes too! I am an International Chess Grandmaster, and (loooooong ago!) I was the World Under-17 Champion. I am the author of five published chess books and hundreds of newspaper articles. I live with my wonderful wife and son in Belgium. I also love music, movies and puzzles. I blog at paulmotwani.com. My e-mail address is pmotwani141@gmail.com. You can find me on Facebook, too.

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