DAY/DYDD 80:

The Main Challenge

Each of the nine letters, A-I, in the two sections below has a subtraction calculation attached to it.  Which is the only letter that has the same answer in both sections?

• Section 1

C:24–13  G:14–6  D:20–10  E:15–7  I:14–9  F:18–11  B:9–2  A:17–5  H:25–12

• Section 2

I:16–3  A:15–1  H:20–8  C:19–9  G:12–4  E:11–5  F:21–6  D:14–7  B:27–18

The 7puzzle Challenge

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

The 4th & 7th rows contain the following fourteen numbers:

3   4   10   11   24   27   30   32   35   44   54   60   70   77

What is the difference between the highest and lowest multiples of 4?

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 just TWO ways of making 80 when using Lagrange’s Theorem. Can you find them both?

Can you arrive at 80 by inserting 4, 5, 8 and 10 into the gaps on each line?

•  ◯×◯+◯×◯ = 80
•  ◯×◯×(◯–◯) = 80
•  ◯×◯×(◯–◯)² = 80
•  (◯–◯)×(◯–◯)² = 80
•  ◯³+◯×◯÷◯ = 80

Answers can be found here.

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