Puzzle 13
What 5-digit number am I thinking of?
It has two prime digits.
Digit 3 is the highest digit.
Digit 2 is the lowest digit.
Digit 1 is higher than the sum of digits 4 and 5.
Digit 5 is half of digit 4.
Digit 1 is one smaller than digit 3.
Digit 5 is between digit 2 and digit 1
The digit 0 doesn't appear, and there are no duplicates. Don't forget that 1 isn't prime, but 2, 3, 5 and 7 are prime.
Puzzle Copyright © Kevin Stone
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Hint
What can digits 4 and 5 be?
Answer
71842.
Reasoning
The number can be represented as:
- - - - -
By (5), Digit 5 is half of Digit 4, so we have four possibilities:
- - - 2 1
- - - 4 2
- - - 6 3
- - - 8 4
By (4), Digit 1 is higher than the sum of Digit 4 and Digit 5, so we can eliminate two possibilities, leaving:
- - - 2 1
- - - 4 2
By (7), Digit 5 is between Digit 2 and Digit 1, so we can eliminate - - - 2 1, leaving:
- - - 4 2
By (4), Digit 1 must be higher than the sum of Digit 4 and Digit 5, so we have three possibilities:
7 - - 4 2
8 - - 4 2
9 - - 4 2
By (6), Digit 1 is 1 smaller than Digit 3 we can eliminate 9 - - 4 2, and fill in Digit 3, leaving:
7 - 8 4 2
8 - 9 4 2
By (3), Digit 2 is the lowest digit, and must be 1:
7 1 8 4 2
8 1 9 4 2
By (1), the only possibility is:
7 1 8 4 2
Puzzle 14
Last week I drove from Aardvark to Beeville.
On the first day, I travelled 1/3 of the starting distance.
On day two, I travelled 1/2 of the remaining distance.
On day three, I travelled 2/3 of the remaining distance.
At the end of day four, after travelling 3/4 of the remaining distance, I was still 5 miles away from Beeville.
How many miles had I travelled so far?
Puzzle Copyright © Kevin Stone
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Hint
Try working backwards.
Answer
175 miles.
The total distance was 180 miles, but as I still had 5 miles to go, the required answer is 175 miles.
Reasoning
Let's try working backwards.
Day 4: I travelled 3/4, which means that the other 1/4 was 5 miles — so 20 miles were left at the start of day four.
Day 3: I travelled 2/3, which means that the other 1/3 was 20 miles — so 60 miles were left at the start of day three.
Day 2: I travelled 1/2, which means that the other 1/2 was 60 miles — so 120 miles were left at the start of day two.
Day 1: I travelled 1/3, which means that the other 2/3 was 120 miles — so 180 miles were ahead at the start of day one.
The full distance was therefore 180 miles, but we were asked how many miles I had travelled so far, and as I still had 5 to go the answer is 175 miles.
Double-Checking
Start distance: 180 miles.
Day 1: I travelled 1/3 of the distance (60 miles), leaving 120 miles.
Day 2: I travelled 1/2 of the distance (60 miles), leaving 60 miles.
Day 3: I travelled 2/3 of the distance (40 miles), leaving 20 miles.
Day 4: I travelled 3/4 of the distance (15 miles), leaving 5 miles.
Puzzle 15
Imagine a chessboard of size 20 x 20 instead of the normal 8 x 8.
How many squares, of any size, are there in total?
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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
Total = 400 + 361 + 324 + 289 + 256 + 225 + 196 + 169 + 144 + 121 + 100 + 81 + 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 2870.
Puzzle 16
My local library decided to hand out some of its old books to the 1,400 people who lived in the town.
To each library member, they offered 6 books, and to each non-member, they offered 4 books.
If only half of the members, and three-quarters of the non-members, accepted the books, how many books were given away?
Puzzle Copyright © Kevin Stone
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Hint
As we don't know how of the people were library members, you can try a random number and see what happens.
Answer
4,200 books.
Reasoning
It doesn't matter how many library members there were, as 1/2 of them accepted the books, and each accepted 6 books, which is 3 books per member on average.
Similarly, it doesn't matter how many non-members there were as 3/4 of them accepted the books, and each accepted 4 books, which is again 3 books per non-member on average.
Double-Checking
Both the members and non-members accepted, on average, 3 books each.
And 1400 x 3 = 4200.
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