# day/dydd 72 at 7puzzleblog.com T he Main Challenge

From the numbers 1 to 20 inclusive, find the only one that remains when all square numbers, multiples of 6, factors of 40 and odd numbers are eliminated. The 7puzzle Challenge

The playing board of the 7puzzle game is a 7-by-7 grid of 49 different numbers, ranging from up to 84.

The 1st & 6th rows contain the following fourteen numbers:

2   5   9   12   14   15   18   20   22   33   40   49   56   72

How many square numbers are present? The Lagrange Challenge

Lagrange’s Four-Square Theorem states that every positive integer can be made by adding up to four square numbers.

For example, 7 can be made by 2²+1²+1²+1² (or 4+1+1+1).

There are THREE ways of making 72 when using Lagrange’s Theorem. Can you find them? The Mathematically Possible Challenge

Using 46 and 12 once each, with + – × ÷ available, which FOUR numbers is it possible to make from the list below?

9    18    27    36    45    54    63    72    81    90

#9TimesTable

The Target Challenge

Can you arrive at 72 by inserting 4, 6, 8 and 10 into the gaps on each line?

•  ◯×◯++ = 72
•  ◯×◯–(◯+√◯) = 72
•  (◯+◯)×+◯ = 72   