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Elastic Numbers

Throughout this puzzle, expressions like $$AB$$ will represent the digits of a number, not $$A$$ multiplied by $$B$$.
A two-digit number $$AB$$ is called elastic if:
1. $$A$$ and $$B$$ are both non-zero.
2. The numbers $$A0B$$, $$A00B$$, $$A000B$$, ... are all divisible by $$AB$$.
There are three elastic numbers. Can you find them?

Square Pairs

Source: Maths Jam
Can you order the integers 1 to 16 so that every pair of adjacent numbers adds to a square number?
For which other numbers $$n$$ is it possible to order the integers 1 to $$n$$ in such a way?

Factorial Pattern

$$1\times1!=2!-1$$ $$1\times1!+2\times2!=3!-1$$ $$1\times1!+2\times2!+3\times3!=4!-1$$
Does this pattern continue?

24 December

Today's number is 191 more than one of the other answers and 100 less than another of the answers.
Tags: numbers

23 December

Today's number is the number of three digit numbers that are not three more than a multiple of 7.
Tags: numbers

22 December

Today's number is a palindrome. Today's number is also the number of palindromes between 111 and 11111 (including 111 and 11111).

19 December

The sum of all the numbers in the eighth row of Pascal's triangle.
Clarification: I am starting the counting of rows from 1, not 0. So (1) is the 1st row, (1 1) is the 2nd row, (1 2 1) is the 3rd row, etc.

18 December

The smallest number whose sum of digits is 25.