Dear Readers,
I am grateful for each moment of every day and, regarding the future, I know that it is always in God’s hands,
but tonight I do have some fun mathematical surprises that I hope to be sharing with more friends tomorrow…One’s life begins, of course, long before the date of birth, but for simplicity, if I count my date of birth (Wednesday 13 June 1962) as my day number 1, then tomorrow (Monday 25 January 2021) will be day number 21412 for me.
I find 21412 to be a rather beautiful palindrome containing the number 141 which has been special on many occasions throughout my life, and since 21 x 4=84 I am all the more happy to be writing about this now in Blog Post #84 !
Earlier today, I was thinking specifically about the facts that 84=7 x 12 too, and also 6 x 14…the 7 went down by one to 6, while the 12 went up by two to 14. I wondered which other positive whole numbers could, just like 84, be expressed in the form a x b and also (a-1) x (b+2).
I found 4=2 x 2 = 1 x 4; 12 = 3 x 4 = 2 x 6; 24 = 4 x 6 = 3 x 8; 40 = 5 x 8 = 4 x 10; 60 = 6 x 10 = 5 x 12; and the largest two-digit solution, our 84 = 7 x 12 = 6 x 14.
It’s not surprising that all the solutions are even numbers (more than that, they must be multiples of 4) because they are all actually of the form 2 x a x (a-1) since, starting from the required condition a x b = (a-1) x (b+2), we can show (using just a little algebraic simplification) that b must equal 2 x (a-1).
The expression (a-1) x a x 2 makes me think of my lovely youngest sister who is about to celebrate a special birthday for which I can say, “If you multiply her whole number age now by the new age that she’s about to be, and then double the result, the final result is a square number“.
That can happen three times in our lives:
(besides the case of 0 x 1 x 2 = 0 = 0 squared)
1 x 2 x 2 = 4 = 2 squared;
8 x 9 x 2 = 144 = 12 squared;
49 x 50 x 2 = 4900 = 70 squared.
Though greater solutions don’t occur within our earthly, natural lives for now…
…looking forward to eternity in Heaven,
there’s 288 x 289 x 2 = 166464 = 408 squared !
Would you like to happily go even higher…!?
Personally, I’m 58 now and always still grateful for every moment here, but I would also love to spend eternity in Heaven when it’s time.
In 1682, Edmond Halley observed Halley’s Comet and correctly predicted that it would light up our skies beautifully about every three-quarters of a century, so I’ll round off with this wee tribute tonight:
1681 x 1682 x 2 = 5654884 = 2378 squared,
and we can also get exactly the same result from
((141-100) x 58) squared !
I look forward hopefully to sharing further fun surprises with you soon!
With very best wishes as always,
Paul Motwani xxx
Paul Motwani 