Blog Post #93: Happy Birthday, Kristina!

Dear Readers,

Today, February 20, is the 20th birthday of Kristina, a super-talented violinist who is also very good at Mathematics.

Let’s warm up with the following steps in honour of Kristina…

1. Start with 93, the ordinal number of this blog post.
2. Add on 100, for Kristina’s magical performances on the violin.
3. Multiply by my favourite number, 3.
4. Since I love it so much, let’s repeat step number 3 !!

Kristina wouldn’t need a calculator to figure out (93 + 100) x 3 x 3 = 1737,

and I’m pretty sure the result would make her laugh again–at least a little bit!–as I’ve often cooked up quick, fun puzzles resulting in the number 1737, because Kristina’s treasured violin dates way back to that very year!

Now it’s time for a couple of fun brainteasers for Kristina’s 20th birthday!

Start by considering 1 + 2 + 3 + 2 + 1 = 9

&  1 + 2 + 3 + 4 + 3 + 2 + 1 = 16.

Can you now state the result of 1 + 2 + 3 + … + 19 + 20 + 19 + … + 3 + 2 + 1 ?

I’m now thinking of another ‘secret whole number’. Let’s call it K.

Special K-puzzle

The sum 1 + 2 + 3 + … + K + … + 3 + 2 + 1

gives a result beginning with the digits 20… on the left, followed by more digits.

What is the smallest possible value for K that would produce such a result? 

I’ll let you and Kristina and her friends enjoy figuring out the brainteaser answers, and solutions will (God-willing as always) be posted at the time of the next blog post.

Puzzle Solutions (being posted now after Kristina has already quickly cracked the puzzles on her own birthday! Congratulations!)

Kristina spotted that in 1 + 2 + 3 + 2 + 1 = 9, the result is equivalent to 3 squared, and similarly in 1 + 2 + 3 + 4 + 3 + 2 + 1 = 16, the result is equivalent to 4 squared. By analogy (though the result can, of course, be proved rigorously if needed), 1 + 2 + 3 + … + 19 + 20 + 19 + … + 3 + 2 + 1 = 20 squared which is 400.

Continuing that theme, 1 + 2 + 3 + … + K + … + 3 + 2 + 1 = K squared.

Now, the two-digit number 20 is NOT a square number.

Also, since 14 squared = 196 and 15 squared = 225, in-between there can’t be any three-digit whole square number beginning with 20…

However, 44 squared = 1936 and 45 squared = 2025.

So, 45 is the smallest possible value for K which gives the desired result, beginning with 20… after squaring.

K = 45.

Wishing everyone a wonderful weekend now,

Paul Motwani xxx