During the recent BrainBashers Football Competition, 5 teams were competing.
Each team played each other just once, with 2 points for a win, 1 point for a draw, and 0 points for a loss.
However, the rain damaged the final scoresheet, and this is all that could be read of the final points after all of the matches:
Rugby 6
Grimsby 5
Hereford 4
Lichfield 2
Reading ?
Can you work out how many points Reading ended the season with?
Reasoning
There were 5 teams in the league, and every team played every other, so there were a total of 10 games played:
Rugby vs Grimsby
Rugby vs Hereford
Rugby vs Lichfield
Rugby vs Reading
Grimsby vs Hereford
Grimsby vs Lichfield
Grimsby vs Reading
Hereford vs Lichfield
Hereford vs Reading
Lichfield vs Reading
We know that 2 points were allocated for each game played. Which means that there were a total of 20 points over the 10 games.
Since the given results add to 17, the remaining 3 points belong to Reading.
?
Puzzle 10
Last week, I travelled from London to Stoke.
On the first day, I travelled one half of the distance.
On day two, I travelled one third of the remaining distance.
On day three, I travelled three quarters of the remaining distance.
Yesterday I travelled one half of the remaining distance.
Reasoning
I now have 5 miles left to travel, so I must have travelled 5 miles yesterday.
At the end of Day 3, I had 10 miles left to travel, so I must have had 40 miles left at the start of the Day 3.
At the end of Day 2, I had 40 miles left to travel, so I must have had 60 miles left at the start of the Day 2.
At the end of Day 1, I had 60 miles left to travel, so I must have had 120 miles left at the start of the Day 1.
Double-Checking
On the Day 1, I travelled 60 miles, leaving 60 miles.
On Day 2, I travelled 20 miles, leaving 40 miles.
On Day 3, I travelled 30 miles, leaving 10 miles.
Yesterday, I travelled 5 miles, leaving 5 miles.
??
Puzzle 11
Can you find a five-digit number that has no zeros, no ones, no digit is repeated, and:
the fourth digit is a quarter of the total of all of the digitsthe second digit is twice the first digitthe third digit is the largestthe last digit is the sum of the first two digits
Reasoning
We can start by labelling the digits as ABCDE.
We know that:
(i) B = 2 x A
and:
E = A + B
And using (i) we get:
E = A + (2 x A) (ii) E = 3 x A
If A = 1, this isn't allowed (as there are no 1's in the puzzle).
If A = 2, then B = 4, and E = 6.
If A = 3, then B = 6, and E = 9, but this isn't allowed (as C has to be the largest digit).
So, A = 2, B = 4, E = 6, and we now have to find C and D.
We also know that:
D = (A + B + C + D + E) ÷ 4
And using (i) and (ii) we get:
D = [A + (2 x A) + C + D + (3 x A)] ÷ 4
so:
3 x D = (6 x A) + C
so:
(iii) D = [(6 x A) + C] ÷ 3
C can only be 7, 8 or 9 (as it's the largest digit, and we've already found 6) and (iii) tells us that it must be a multiple of 3, which means that C = 9. Leaving D = 7.
So the final number is: 24976.
Double-Checking
The answer is 24976.
The fourth digit is a quarter of the total of all of the digits.
A + B + C + D + E = 2 + 4 + 9 + 7 + 6 = 28, and 28 ÷ 4 = 7.
The second digit is twice the first digit.
4 = 2 x 2.
The third digit is the largest.
9 is the largest digit.
The last digit is the sum of the first two digits.
6 = 2 + 4.
??
Puzzle 12
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, as the prime numbers start with 2, 3, 5, …
