# DAY/DYDD 135: T he Main Challenge

Here’s a number trail involving 16 arithmetical steps.  Start with the number 2, then:

•  +33
•  divide by seven
•  ×3
•  double this
•  find the square root of this
•  +7
•  1/2 of this
•  subtract two
•  ÷6
•  +15
•  ×5
•  –93
•  ÷2 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 3rd & 6th rows contain the following fourteen numbers:

5   12   13   18   20   25   33   36   42   45   49   56   66   80

Which two numbers, when doubled, 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 135 when using Lagrange’s Theorem. Can you find them? The Mathematically Possible Challenge

Using 36 and 10 once each, with + – × ÷ available, which THREE numbers is it possible to make from the list below?

10    20    30    40    50    60    70    80    90    100

#10TimesTable

The Target Challenge

Can you arrive at 135 by inserting 4, 5, 9 and 12 into the gaps on each line?

•  ◯²–◯×(◯–◯) = 135
•  ◯×◯×◯÷◯ = 135
•  (◯+◯)²+half(◯×◯) = 135   