Random Puzzles 

Workings

Hint
There are no squares of size 5 x 5.

Answer
There are 125 squares without a rook.

Reasoning
61 squares of size 1 x 1
39 squares of size 2 x 2
20 squares of size 3 x 3
 5 squares of size 4 x 4

Giving a total of 61 + 39 + 20 + 5 = 125 squares.

Workings

Hint
There is more than one way of completing this in 4 moves.

Answer
Ball bell hell heal head.
Ball bell hell held head.
Ball hall hell heal head.
Ball hall hell held head.

Workings

Hint
The answer begins with 3.

Answer
64 x 58 = 3712.

Reasoning
Since no digit is duplicated, neither number can end in 1, otherwise, the last digit of the answer would already have been used.

Neither number can end in 5 because the answer would then end in 5 or 0.

So the first number can only be 62, 63, 64, 67, or 68.

We can now look at what the second number can end with, and we find that …

   if the first number was 62, the second number can only end in 4 or 7.

Why …
   not 1 as previously explained
   not 2 because we've already used that in the 62
   not 3 because 62 x *3 would end in 6, which is already in the 62
   not 5 as previously explained
   not 6 because we've already used that in the 62
   not 8 because 62 x *8 would end in 6, which is already in the 62

We can repeat this for the other possible first numbers and find that …

   if the first number was 62, the second number can only end in 4 or 7.
   if the first number was 63, the second number can only end in 4, 7, or 8.
   if the first number was 64, the second number can only end in 2, 3, 7, or 8.
   if the first number was 67, the second number can only end in 2, 3, or 4.
   if the first number was 68, the second number can only end in 3 or 4.

Let's check these in turn …

If the first number was 62, the only possible values are:

   62 x 14 = 868
   62 x 34 = 2108
   62 x 54 = 3348
   62 x 74 = 4588

or

   62 x 17 = 1054
   62 x 37 = 2294
   62 x 57 = 3534
   62 x 87 = 5394

Only 4 calculations are required for each option, as we didn't need to check 62 x 84 as we already know that the answer would end in 8, or 62 x 47 as we already know that the answer would end in 4.

All of these answers fail as they don't contain all of the digits.

Similarly, we can look at 63, then 64, etc. For each of the numbers, we have to check the possible endings, but in each case, only 4 calculations are required (as seen in the examples above).

We (quickly?) find 64 x 58 = 3712.

Workings

Hint
Cockroaches have 6 feet, mice have 4, and birds have 2. They all have one head!

Answer
35 birds, 5 mice, and 10 cockroaches.

Reasoning
Cockroaches have 6 feet, mice have 4, and birds have 2. They each have one head.

We can write down expressions for the heads and the feet, calling birds B, mice M, and cockroaches C.

Counting the heads:
(1)    B +  M +  C =  50

Counting the feet:
(2)   2B + 4M + 6C = 150

We are told that for every mouse there are two cockroaches so, C = 2M. Update (1) and (2) to give:
(3)    B +  3M =  50
(4)   2B + 16M = 150

If we double (3) we get:
(5)   2B +  6M = 100

We can now do (4) − (5) to give:
 10M = 50
   M = 5

So, we have 5 mice (and 10 cockroaches).

We can use M = 5 in (3) to give:
 B + 3 x 5 = 50
    B + 15 = 50
         B = 35

So, C = 10, M = 5 and B = 35.

Double-Checking
10 cockroaches have 10 heads and 60 feet.
5 mice have 5 heads and 20 feet.
35 birds have 35 heads and 70 feet.

The total number of heads = 10 + 5 + 35 = 50.

The total number of feet = 60 + 20 + 70 = 150.