Hard Puzzles 

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
Which 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
The largest possible difference is when 1 is next to 8, a difference of 7. Since we have 7 differences to find, and the largest possible difference is 7, all of the possible differences must exist. Start by looking where can the 8 go.

Answer
 


Reasoning
The largest possible difference is when 1 is next to 8, a difference of 7. Since we have 7 differences to find, and the largest possible difference is 7, all of the possible differences must exist: 1, 2, 3, 4, 5, 6, 7, and let's call these D1, …, D7.

D7 can only happen when:
   1 is next to 8 = D7

D6 can happen when:
   1 is next to 7, but these are given numbers that are not next to each other.
   2 is next to 8 = D6

Where can 8 go? If we put 8 above 1, we cannot also satisfy D6 (2 is next to 8).

Therefore, we have two possibilities:
   (a) 8 to the left of 1
   (b) 8 to the right of 1
(a) 8 to the left of 1

By D6, 2 would be below 8, and this would give us D1, D6, D7. What can we place to the right of 1?



3 – no, because the difference between 1 and 3, and the difference between 3 and 5, are both D2.
4 – no, because the difference between 4 and 5 is D1, which we would already have.
6 – no, because the difference between 5 and 6 is D1, which we would already have.

There are no possible numbers we can place to the right of 1, so 8 can't go to the left of 1.
(b) 8 to the right of 1

By D6, 2 would be above 8, and this would give us D3, D6, D7.



4 can't go next to 1, otherwise we'd create another D3. Therefore, 4 goes in the bottom left corner.



We are now left with 3 and 6.

If 6 went above 4, and 3 above 1, these would both be D2.

Therefore, 3 goes above 4, 6 goes above 1.

The final answer is:

Workings

Hint
The two left numbers multiplied, or the right two numbers, should create the central number.

Answer
The way to arrange the sacks of flour is as follows: 2, 78, 156, 39, 4. Here each pair when multiplied by its single neighbour makes the number in the middle, and only five of the sacks need to be moved.

There are just three other ways in which they might have been arranged (4, 39, 156, 78, 2; or 3, 58, 174, 29, 6; or 6, 29, 174, 58, 3), but they all require the moving of seven sacks.