Dear Readers,
It doesn’t really matter whether ‘colors’ includes a ‘u’ or not, but the unique colors and all the love that you personally can add to the world by being you, yes, that always matters to someone; you can be happy and sure about that.
The caption with a photo that I saw a few days ago was ‘Everyone smiles in the same language’, and so let’s add some more lovely smiles in our lives to further brighten them for others, too.
Sincere special thanks to Paul Fitzpatrick
for many gems of advice about chess, teaching, love and life.
Near the end of the delightful 1959 Walt Disney Productions short film, ‘Donald in Mathmagic Land’, Donald Duck encounters a seemingly infinite row of doors that are locked because behind them are wonderful surprises waiting to be discovered by curious minds.
I was reminded yesterday of Donald’s adventures when I received the following colorful photo from England…
Imagine that the red, blue, yellow, green, red, blue, yellow, green,…doors (repeating in an infinite row of colors, starting with red at the very beginning) are numbered 1, 2, 3, 4, 5, 6, 7, 8,…
Mentally walk along and choose any four consecutive door numbers that you like.
Multiply your four consecutive door numbers together, and note the result of that product. Let’s call your product number result P.
THREE FUN QUESTIONS TO PONDER WITH DONALD 😊😊😊
- What is the color of door number P ?
- For everyone, P is guaranteed to always be a multiple of a certain number M that I think of every day. What is the maximum value that I can always correctly claim for M ?
- For everyone, P + A is guaranteed to always be a square number. What particular number is A ?
I’m hoping that some colleagues and some students at the beautiful Musica Mundi School (where I love working as the Mathematics Teacher) will happily and quickly send me their good answers! Then I will have the pleasure of awarding some nice prizes on Friday morning this week, before the start of our school holiday break.
😊😊😊
While you have fun cracking the puzzles with Donald, please also enjoy another gorgeous photo (that I received yesterday) from England. I didn’t want to duck out of sending it; it’s an absolute quacker!!
Wishing you and everyone a wonderful, happy day now,
With love,
Paul Motwani xxx
P.S. I intend to publish solutions to the puzzles in a few days’ time.
Here they are now, on Monday 1 November 2021:-
- First, recall that, in the repeating color sequence of red, blue, yellow, green doors, every 4th door was a green one. Since any choice of 4 consecutive whole numbers must include a multiple of 4, the product P will definitely be a multiple of 4, and so door number P must always be a green one.
- If you were to choose consecutive door numbers 1, 2, 3 and 4, their product would just be 24. So, our highest possible claim is that the product P might always a multiple of 24. We cannot claim more, but is that claim about P being a multiple of 24 guaranteed to always be true? Happily, it is, because any choice of 4 consecutive whole numbers must include a multiple of 2, a multiple of 3, and a multiple of 4, which guarantees that the product will always be a multiple of 2 x 3 x 4 i.e. a multiple of 24. Therefore, M = 24.
- If we let n stand for the smallest number amongst four consecutive whole numbers chosen, then the product P = n(n+1)(n+2)(n+3). It follow that, if A = 1, P + A = n(n+1)(n+2)(n+3) + 1, which can be shown to be exactly equivalent to (n squared + 3n + 1) squared, always a perfect square.
For example, if n = 1 and A = 1, then P + A = 1 x 2 x 3 x 4 + 1 = 25 = 5 squared;
if n = 2 and A = 1, then P + A = 2 x 3 x 4 x 5 + 1 = 121 = 11 squared;
if n = 3 and A = 1, then P + A = 3 x 4 x 5 x 6 + 1 = 361 = 19 squared, and so on.
With A = 1, P + A is always a perfect square.
Many congratulations to 12-year-old Wout Callens who solved all the puzzles in style! Wout was presented with a nice book prize and some delicious sweets at Musica Mundi School this past Friday. 😊😊😊
Paul Motwani 