# day/dydd 137 at 7puzzleblog.com

T he Main Challenge

It is possible to use seven 5’s (5 5 5 5 5 5 and 5) once each, with the four operations + – × ÷, to make all the target numbers from 1 to 5.

For instance, to arrive at the target numbers 1 and 2, you can do:

• [(5+5)÷5 – 5÷5] × 5÷5 = 1
• [(5+5)÷5 – 5÷5] + 5÷5 = 2

Your task is to show how to arrive at the other target numbers 3, 4 and 5.

The 7puzzle Challenge

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

The 5th & 7th rows contain the following fourteen numbers:

4   6   7   11   16   21   24   27   30   50   70   77   81   84

What is the product of the two prime numbers listed?

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 SIX ways of making 137 when using Lagrange’s Theorem. Can you find them?

The Mathematically Possible Challenge

Using 36 and 10 once each, with + – × ÷ available, which TWO numbers is it possible to make from the list below?

1    4    9    16    25    36    49    64    81    100

#SquareNumbers

The Target Challenge

Can you arrive at 137 by inserting 4, 5, 6 and 8 into the gaps on each line?

•  ◯³+◯²+◯×◯ = 137
•  ◯³+◯×◯÷◯ = 137

Answers can be found here.

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