Hint
How many 1 x 1 squares are there, how many 2 x 2, …?
Answer
2,870 squares.
Reasoning
We have squares sizes from 1 x 1 up to 20 x 20, and these are the counts:
1 x 1 = 400
2 x 2 = 361
3 x 3 = 324
4 x 4 = 289
5 x 5 = 256
6 x 6 = 225
7 x 7 = 196
8 x 8 = 169
9 x 9 = 144
10 x 10 = 121
11 x 11 = 100
12 x 12 = 81
13 x 13 = 64
14 x 14 = 49
15 x 15 = 36
16 x 16 = 25
17 x 17 = 16
18 x 18 = 9
19 x 19 = 4
20 x 20 = 1
Hint
Look at the letter positions in the alphabet.
Answer
R.
If you take the alphabetic positions of each letter in a row, the total is always 56.
Double-Checking
Y + G + W + A = 25 + 7 + 23 + 1 = 56
K + L + R + O = 11 + 12 + 18 + 15 = 56
Z + J + P + D = 26 + 10 + 16 + 4 = 56
V + F + T + H = 22 + 6 + 20 + 8 = 56
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Puzzle 155
My local shop wants to create a sign that says PUZZLES, and they have the following letters, each in a different colour:
P P P U U Z Z Z Z Z L L L L E S S
How many different ways can they make the sign?
Hint
There are 3 P's, so there are 3 different ways to start the word.
Answer
960.
Reasoning
Given: P P P U U Z Z Z Z Z L L L L E S S.
The order of the Z's is important. For example, we could chose a red one and a green one, so the order they appear in the sign has to be taken into consideration.
There are:
3 different P's that could be chosen.
2 different U's that could be chosen.
5 different Z's that could be chosen.
4 different Z's that could be chosen, as one has already been chosen.
4 different L's that could be chosen.
1 E that could be chosen.
2 different S's that could be chosen.
So there are 3 x 2 x 5 x 4 x 4 x 1 x 2 = 960 different ways to make the sign.
???
Puzzle 156
Below is a very special grid, around each shaded number are 8 white squares.
However, each white square should have a number from 1 to 7.
Once filled in, these 8 numbers will sum to the shaded number.
In addition, once completed correctly, no row nor column will contain a duplicate number within a white square.
For example, the top row may be 5 6 4 2 3 1 7, etc.
This is a very difficult puzzle, and many people resort to using a computer to help them, which is a challenge in itself.
