Hard Puzzles 

Workings

Hint
Look at the number of sides of each of the shapes.

Answer
Square.

Reasoning
Square the first shape's sides, and the remaining shapes add up to this number.
   triangle (3 x 3) = pentagon (5) + square (4)

   square   (4 x 4) = hexagon (6) + hexagon (6) + square (4)

   pentagon (5 x 5) = hexagon (6) + hexagon (6) + hexagon (6) + square (4) + triangle (3)

Therefore:
   hexagon  (6 x 6) = octagon (8) + octagon (8) + octagon (8) + octagon (8) + square (4)

Workings

Hint
These are all Greek letters.

Answer
θ (theta).

Reasoning
These are all Greek letters, and they are:

   xi, chi, theta, psi, pi, phi

They all sound similar to each other, except θ.

Workings

Hint
What letters are missing?

Answer
The letter was written on a typewriter that has a problem with the top row of letters – they did not print correctly. The letter should have read:

Dear Kevin,

I would like to complain about the top row of my typewriter.

These letters do not appear to be working correctly.

As you can see, they are missing from this document.

I would be very grateful if you could correct this problem immediately.

Kind regards.

Sam South.

Workings

Hint
What are the possible differences?

Answer
 


Reasoning
The first observation is that all of the possible gaps must exist: 1, 2, 3, 4, 5, 6, 7, and let's call these Gap1, ..., Gap7.

Looking at Gap7, this can only happen when:
   [Gap7a: 1 is next to 8]

Looking at Gap6, this can happen when:
   [Gap6a: 1 is next to 7]
      this is not allowed because 1 and 7 are given squares that are not next to each other.
   [Gap6b: 2 is next to 8]

Looking at Gap5, this can happen when:
   [Gap5a: 1 is next to 6]
   [Gap5b: 2 is next to 7]
      this is not allowed because [Gap6b: 2 is next to 8], which would mean that we can't also have [Gap7a: 1 is next to 8].
   [Gap5c: 3 is next to 8]
      this is not allowed because we couldn't then also have: [Gap7a: 1 is next to 8] and [Gap6b: 2 is next to 8].

We can't have the 8 above the 1, as there is then no place for [Gap6b: 2 is next to 8].

So, we have two possible places for the 8, to the left of the 1, or to the right of the 1.

8 to the left of 1



What can we place to the right of 1?

3 – no, because the gap between the 1 and 3, and the gap between 3 and 5, are both 2.
4 – no, because the gap between 4 and 5 is 1, which we already have.
6 – no, because the gap between 5 and 6 is 1, which we already have.

So the 8 can't go to the left of 1.

8 to the right of 1

We know [Gap5a: 1 is next to 6].

Let's try 6 to the left of 1. No matter where we place the 3 and 4 we end up with duplicate gaps. So this doesn't work.



So, the 6 must go above 1, but the 4 can't go to the left of 1 because of duplicate gaps. Therefore, this doesn't work either.



Which means that the final answer is: