# DAY/DYDD/PÄIVÄ/NAP 84 T he Main Challenge

1.  Work out the sum of 3 and 5,
2.  Calculate the difference between 3 and 5,
3.  Find the product of the two answers in Points 1 and 2, 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 2nd & 3rd rows contain the following fourteen numbers:

8   13   17   25   28   36   42   45   48   55   63   64   66   80

Which pair of numbers have a difference of 49? 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 84 when using Lagrange’s Theorem. Can you find them? The Mathematically Possible Challenge

Using 23 and 11 once each, with + – × ÷ available, which is the ONLY number that is 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 84 by inserting 2, 3, 6 and 7 into the gaps on each line?

•  (◯+◯)²+◯–◯² = 84
•  (◯×◯+◯)× = 84
•  ((◯–◯)²–◯)×◯ = 84   