# day/dydd 89 at 7puzzleblog.com

T he Main Challenge

Of the numbers 1, 2, 3 … 6000, how many are NOT multiples of 2, 3 or 5?

The 7puzzle Challenge

The playing board of the 7puzzle game is a 7-by-7 grid containing 49 different numbers, ranging from 2 up to 84.

The 4th & 5th rows contain the following fourteen numbers:

3   6   7   10   16   21   32   35   44   50   54   60   81   84

What is the average of the two prime numbers on the list?

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 FIVE ways of making 89 when using Lagrange’s Theorem. Can you find them?

The Mathematically Possible Challenge

Using 23 and 11 once each, with + – × ÷ available, which THREE numbers is it possible to make from the list below?

30    31    32    33    34    35    36    37    38    39

#NumbersIn30s

The Target Challenge

Can you arrive at 89 by inserting 1, 4, 8 and 10 into the gaps on each line?

•  ²+◯²+◯ = 89
•  ◯×◯+(◯–◯)² = 89
•  (◯+◯)³–double(◯+◯) = 89

Answers can be found here.

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