# DAY/DYDD 69: The Main Challenge

If the number sequence  5  9  13  17  21 . . .  is continued, which is the only number from the following list that will appear later in the sequence?

35    43    59    67    71    85    95 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 3rd & 5th rows contain the following fourteen numbers:

6   7   13   16   21   25   36   42   45   50   66   80   81   84

Is the sum of the even numbers more than double the sum of the odd numbers? 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 FOUR ways of making 69 when using Lagrange’s Theorem. Can you find them? The Mathematically Possible Challenge

Using 46 and 12 once each, with + – × ÷ available, which TWO numbers are NOT possible to make from the list below?

6    12    18    24    30    36    42    48    54    60

#6TimesTable

The Target Challenge

Can you arrive at 69 by inserting 2, 3, 4 and 7 into the gaps on each line?

•  ◯²×◯+◯+ = 69
•  ◯³+(◯+◯)÷ = 69
•  (◯²+◯)×–double◯ = 69   