Blog Post #74: For Genevieve, Julie, Marcello, Timothee and Yordi!!

Dear Readers,

This post is just in time to wish Genevieve a very happy birthday tomorrow, while it’s exactly nine weeks early for the birthday of Genevieve’s mum, Julie, a lovely lady I know through my Mathematics teaching. A bright student named Timothee is a keen chess player, as are ‘Mathematical Marcello’ (a boy whom I remember well from my previous school, especially when as a little 8-year-old then, he mentally calculated 99×99=9801 in just a couple of seconds!) and Yordi, who is a very good friend of my son.

Photo by Christophe Gillain.

Please enjoy today’s Maths, Chess and Word puzzles dedicated to Genevieve, Julie, Marcello, Timothee and Yordi.

Solutions are given at the end of this article.

Genevieve & Julie’s Word Puzzle

Rearrange the letters of RATE VIP or VET PAIR to make a proper 007-letter English word!

Genevieve & Julie’s Birthday Brainteaser!

Given that Julie is 36 years older than her daughter Genevieve, can there be any year(s) in their lives for which the product of Genevieve’s whole number age and Julie’s whole number age in the same calendar year equals a positive square number? For instance, 1×37=37, 2×38=76, 3×39=117,  none of which produce square number results.

Your special challenge is either to prove that no year works or to find every case that does work!

No Ordinary Word Puzzle

Rearrange the letters of YORDI RAN to make a proper eight-letter English word.

Chess Study from 2⁷ years ago!

It’s White to play and win in an elegant and very practical 1892 chess study by Henry Otten.

Wishing you all lots of fun and a wonderful weekend now too,

Paul Motwani xxx

P.S.=Puzzle Solutions!

RATE VIP or VET PAIR can be rearranged to make PRIVATE.

YORDI RAN rearranges to make ORDINARY, but Yordi is no ordinary chess player; he’s actually very talented!

In the chess study, White wins with 1 a5 Bf8 2 Kd5 (preventing …Bc5) 2…Bh6 (aiming for …Be3) 3 g5+!! Bxg5 4 Ke4! (see diagram below) 4…Bh4 5 Kf3! and Black’s bishop cannot prevent the white pawn’s imminent advance a6-a7-a8 to promotion.

Now for the solution to Genevieve & Julie’s Birthday Brainteaser…!

Let Genevieve and Julie’s ages (in years) be the whole numbers x and x+36, respectively.

Their product is x(x+36) which expands to x²+36x.

If that expression could equal a square number, it would clearly be greater than x² but less than (x+18)² because that’s equivalent to x²+36x+324.

Consider (x+17)², which is equivalent to x²+34x+289.

In our problem, it’s not possible that x²+36x=x²+34x+289 because that would reduce to 2x (an even number) having to equal the odd number 289, and so we have a contradiction there. Similar contradictions occur if we try to match x²+36x to (x+any odd number)².

Therefore, to complete our investigation in this puzzle, we only need to examine at most the cases

x²+36x=(x+an even number between 2 to 16 inclusive)².

Remembering that x needs to be a whole number, it turns out that that only happens in two cases:

x²+36x=(x+12)², leading to x=12

and

x²+36x=(x+16)², leading to x=64.

Genevieve was 12 years old some years ago, but I very much hope that (many, many years from now!) Genevieve and Julie will be celebrating birthdays 64 and 100 respectively!!