DAY/DYDD 98:

The Main Challenge

There are two rules to consider when calculating the NEXT number in our special sequence:

•  if it’s EVEN, halve it
•  if it’s ODD, add 1

For example, if starting at 50, the sequence would be:

•   50    25    26    13    14    7    8    4    2    1

So it takes 9 steps to get down from 50 to 1.

Following the above rules, which starting number between 51 and 100 creates the most number of steps (13 steps in all) when eventually arriving at 1?

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 1st & 3rd rows contain the following fourteen numbers:

2   9   13   14   15   22   25   36   40   42   45   66   72   80

Which three numbers, when trebled, are also present 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 SEVEN ways of making 98 when using Lagrange’s Theorem. Can you find them all?

Can you arrive at 98 by inserting 2, 4, 5 and 10 into the gaps on each line?

•  (◯+◯)×(◯+◯) = 98
•  ◯²–◯×(◯–◯) = 98
•  ◯³+double(◯+◯+◯) = 98

Answers can be found here.

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