The triangular palindrome developed in Number Curiosities #1 can be expressed in an alternative arrangement.

Consider the top row in the triangular palindrome. The number 1.

This can be expressed as the fraction:

triangular-palindromes-secret-e1

The next row is the number 121.

This can be expressed as the fraction:

triangular-palindromes-secret-e2

Can you see any pattern forming?

Click to see solution..

Solution

The numerator is the middle digit, (i.e. 2) repeated by as many, (i.e. 22) then squared, (i.e. 22 x 22).

The denominator is simply the sum of the digits, (i.e. 1 + 2 + 1).

See if you can determine how we can express the number in the third row: 12321

Now, see if you can now express each row in the triangular palindrome in this alternative arrangement.

Now, for one, final twist!

Let’s take a closer look at the denominator of this alternative arrangement.

Firstly: 1 + 2 + 1 = 4 = 22

Can we simply the other denominators in a similar fashion? Have a try.

Click to see solution..

Solution

triangular-palindromes-secret-2

Indeed it turns out that the denominators can be expressed as perfect squares.