# day/dydd 20 at 7puzzleblog.com

The Main Challenge

Consider all whole numbers from 1 to 60, then delete the following:

•  all prime numbers,
•  … and any number that differs by 1 from a prime,
•  all square numbers,
•  … and any number that differs by 1 from a square,
•  all multiples of 5,
•  … and any number that differs by 1 from a multiple of 5,
•  all multiples of 7,
•  … and any number that differs by 1 from a multiple of 7.

One number will remain, what is it?

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 1st & 7th rows contain the following fourteen numbers:

2   4   9   11   14   15   22   24   27   30   40   70   72   77

What is the difference between the highest prime number and highest square number?

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 TWO ways of making 20 when using Lagrange’s Theorem. Can you find both?

The Mathematically Possible Challenge

Using 56 and once each, with + – × ÷ available, which are the only TWO numbers it is possible to make from the list below?

11    22    33    44    55    66    77    88    99    110

#11TimesTable

The Target Challenge

Can you arrive at 20 by inserting 1, 4, 6 and 8 into the gaps on each line?

•  (◯–◯)×(◯–◯) = 20
•  (◯÷◯+◯)×◯ = 20
•  (◯+◯)×◯–◯ = 20

Answers can be found here.

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