To mark her 55th birthday, Martha, a school teacher, gave each of her nine pupils a sheet of paper with a single different digit from 1 to 9 written on it.

They stood at the front of the classroom in a row and the 9-digit number on display was divisible by 55. Martha then asked the first 3 in the row (from the left) to sit down. The remaining 6-digit number was also divisible by 55. The next 3 then sat down and the remaining 3-digit number was also divisible by 55.

The 9 digit number was the smallest possible. What was it?

]]>Adam and Eve have convex pentagonal gardens consisting of a square lawn and paving. Both gardens have more than one right-angled corner. All the sides of the gardens and lawns are the same whole number length in metres, but Adam’s garden has a larger total area. Eve has worked out that the difference in the areas of the gardens multiplied by the sum of the paved areas (both in square metres) is a five-digit number with five different digits.

What is that number?

]]>The above is the size of the evil multitude in the last “Infernal indices” novel “A deficit of daemons”.

“Spoiler alert” — at the end, the forces of good engage in the fewest simultaneous battles that prevent this evil horde splitting, wholly, into equal-sized armies for that number of battles. The entire evil horde is split into one army per battle, all equally populated, bar one which has a deficit of daemons, leading to discord and a telling advantage for the forces of good.

How many battles were there?

]]>Oliver lives with his parents, Mike and Nellie, at number 5. In each of numbers 1 to 4 lives a family, like his, with a mother, a father and one child. He tries listing the families in alphabetical order and produces the table below.

House Number |
1_______ |
2_______ |
3_______ |
4_______ |
5_______ |

Fathers |
Alan | Dave | George | John | Mike |

Mothers |
Beth | Ellen | Helen | Kate | Nellie |

Children |
Carole | Freddie | Ingrid | Larry | Oliver |

_______

However, apart from his own family, there is just one father, one mother and one child in the correct position. Neither Helen nor Beth lives at number 3 and neither Dave nor Ingrid lives at number 1. George’s house number is one less than Larry’s and Beth’s house number is one less than Carol’s.

Apart from Oliver’s family, who are correctly positioned?

]]>George and Martha have a set of a dozen balls, identical in appearance but each has been assigned a letter of the alphabet, A, B….K, L and each is made of a material of varying density so that their weights in pounds are 1, 2…. 11, 12 but in no particular order. They have a balance and the following weighings were conducted:

1) **A** + **C** +** I ** v **G** + **J** + **L**

2) **A** + **H** + **L** v **G** + **I** + **K**

3) **B** + **I** + **J** v **C** + **F** + **G**

4) **B** + **D** + **I** v **E** + **G** + **H**

5) **D** + **F** + **L ** v **E** + **G** + **K**

On all five occasions, there was perfect balance and the total of the threesome in each was a different prime number, that in 1) being the smallest and progressing to that in 5) which was the largest.

In alphabetical order, what were the weights of each of the twelve balls?

]]>In the Premier League table a team’s points are usually roughly equal to their goals scored (Burnley were an interesting exception in 2017-18). That was exactly the case in our football league after the four teams had played each other once, with 3 points for a win and 1 for a draw.

A ended up with the most points, followed by B, C and D in that order. Fifteen goals had been scored in total, and all the games had different scores. The best game finished 5-0, and the game BvD had fewer than three goals.

What were the results of B’s three games (in the order BvA, BvC, BvD)?

]]>My wife and I play bridge with Richard and Linda. The 52 cards are shared equally among the four of us — the order of cards within each player’s hand is irrelevant but it does matter which player gets which cards. Recently, seated in our fixed order around the table, we were discussing the number of different combinations of cards possible and we calculated that it is more than the number of seconds since the “big bang”!

We also play another similar game with them, using a special pack with fewer cards than in the standard pack — again with each player getting a quarter of the cards. Linda calculated the number of possible combinations of the cards for this game and she noticed that it was equal to the number of seconds in a whole number of days.

How many cards are there in our special pack?

]]>A tractor is ploughing a furrow in a straight line. It starts from rest and accelerates at a constant rate, taking a two-digit number of minutes to reach its maximum speed of a two-digit number of metres per minute. It has, so far, covered a three-digit number of metres. It now maintains that maximum speed for a further single-digit number of minutes and covers a further two-digit number of metres. It then decelerates to rest at the same rate as the acceleration. I have thus far mentioned ten digits and all of them are different.

What is the total distance covered?

]]>Two teams, A and B, play each other.

A bookmaker was giving odds of 8-5 on a win for team A, 6-5 on a win for team B and X-Y for a draw (odds of X-Y mean that if £Y is bet and the bet is successful then £(X + Y) is returned to the punter). I don’t remember the values of X and Y, but I know that they were whole numbers less than 20.

Unusually, the bookmaker miscalculated! I found that I was able to make bets of whole numbers of pounds on all three results and guarantee a profit of precisely 1 pound.

What were the odds for a draw, and how much did I bet on that result?

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