# day/dydd 180 at 7puzzleblog.com T he Main Challenge

This is a continuation of a number puzzle, posted on DAY 10, made famous by French writer, George Perec.

In that challenge, we asked you to use seven 7’s (7, 7, 7, 7, 7, 7 and 7) once each, with + – × and ÷ available, to make target numbers from 1 to 3.

To continue the puzzle, and make 4, you could simply do:

•  7 – 7÷7 – 7÷7 – 7÷7  =  4

We now invite you to make all the target numbers from 5 through to 9. 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

Which multiple of 8, when 7 is subtracted from it, becomes a square number? The Lagrange Challenge

Lagrange’s Four-Square Theorem states that every 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).

Show how you can make 180, in TEN different ways, when using Lagrange’s Theorem. The Mathematically Possible Challenge

Using 36 and 12 once each, with + – × ÷ available, which is the ONLY target number it is possible to make from the list below?

8    16    24    32    40    48    56    64    72    80

#8TimesTable The Target Challenge

Can you arrive at 180 by inserting 4, 6, 8 and 10 into the gaps on each line?

•  (◯+◯+◯)×◯ = 180
•  (◯+◯)×(◯+◯) = 180
•  ◯²+◯×(◯+◯) = 180
•  (◯+◯+double◯)×◯ = 180   