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# APR Teasers

On this page we will be adding some quiz questions that we post in our monthly client updates - every time we post a new teaser we will update the previous one with a solution, so look out for these each month.

### July 2023 - A Ghastly take on Schrodinger's Cat

Teaser

A king is having a party in 24 hours.

He has 8 casks of wine in the cellar, but someone has informed him that exactly one of them is poisoned.

The poison will kill whoever drinks it sometime, randomly, within 24 hours.

The king is insistent that he won't waste any wine that he doesn't have to, and he doesn't mind sacrificing some of his servants to make sure that he only has to throw out the one poisoned cask.

What is the fewest number of servants the king will need to find out with certainty which cask is poisoned in time for the party?

### May 2023 - A Technical Knock-out

Teaser

A knockout tournament has 64 players. In each round, the remaining players are paired up randomly. The winners progress to the next round and the losers are eliminated. Assume each player has a different skill level and that better players will always beat worse players. How many different finals are possible?

### March 2023 - Nines in 1 to 1,000,000

Teaser

If you were to write every number from one to one million, how many times would you write the number nine?

### February 2023 - A Game of Dice

Teaser

You roll a fair 6-sided dice twice and take the maximum of the two rolls. What is the expected value of this result? For example, if you rolled a 6 and a 3, your result here would be a 6.

### November 2022 - A Meeting at the Train Station

Teaser

Alice and Bob both go to the railway station between 10 and 11am. Each of them turns up at a uniformly-selected, random time between 10 and 11. They then wait for 10 minutes and leave.

What is the probability that Alice and Bob meet each other at the station? (You may assume they meet if there is any point when they are both at the station).

### September 2022 - Cutting a Straight Line

Teaser

If a straight line is cut independently and uniformly at random in two places, what is the probability that the three segments can form a triangle?

### July 2022 -

Teaser

You drop a circular hoop of diameter d<1 onto an infinite chessboard with squares of side length 1.

What is the probability that the hoop surrounds an area containing just one colour?

### June 2022 -

Teaser

What is the smallest positive integer that is 1 more than a multiple of 2, 2 more than a multiple of 3, 3 more than a multiple of 5 and 4 more than a multiple of 7?

### April 2022 - Jiggly Numbers

Teaser

A Jiggly number is a 4-digit number, no digit of which is zero, such that however the digits of the number are rearranged, the resulting 4-digit number is always a multiple of 12.

How many Jiggly numbers are there?

### March 2022 - Avoiding Awkward Introductions

Teaser

There is a round table with 8 chairs. On each chair there is a person.

How many ways are there that each person can shake their right hand with the right hand of somebody else without two pairs of hands crossing each other?

### February 2022 - A Numerical Poem

Teaser

A dozen, a gross and a score,
Plus three times the square root of four,
Divided by seven,
Plus X times eleven,
Is nine squared and not a bit more.

What is x?

### January 2022 - A Strange Extra-Curricular Activity

Teaser
There is a school with 1,000 students and 1,000 lockers. On the first day of term the head teacher asks the first student to go along and open every locker. He asks the second to go to every second locker and close it. He then asks the third to go to every third locker and close it if it is open or open it if it is closed, and so on. The process is completed with the thousandth student.
How many lockers are open at the end?

### December 2021 - Secret Santa

Teaser

APR’s secret Santa consists of 20 people who each write their name on a piece of paper and put it in a hat. Each person randomly draws a name from the hat to determine who has them as their secret Santa.

What is the probability that at least one person draws their own name?

### November 2021 - The Trouble with Time Travel

Teaser

Channelling your inner Marty McFly, you travel one week back in time in order to win the lottery. The jackpot is worth £1 million, and each ticket costs £1. Note that if you win, your ticket purchase is not refunded.

The problem is, you’re not the only person with time-travelling powers. There are 10 other time travellers who also know the winning numbers. You know for a fact that each of them will buy exactly one lottery ticket. Under the lottery’s rules the jackpot is evenly split among all the winning tickets (i.e. not evenly among winning people).

How many tickets should you buy to  maximise your profits, assuming that no non-time-travellers win the lottery?

### October 2021 - APR Mastermind

Teaser

There are 200 episodes in a series of “APR Mastermind”. The first episode of the series features three brand-new contestants. Each subsequent episode includes a returning champion (the winner of the previous episode) as well as two new challengers. Throughout the series, it so happens that the returning champions are particularly strong, with each one winning five consecutive episodes before being dethroned on the sixth.

If you pick a contestant at random from across the series, what is the probability that they are an APR Mastermind champion (meaning they won at least one episode)?

### September 2021 - Exploring a Chess Board

Teaser

A standard 8x8 chessboard has each square marked with a different numeric value according to the following rule: for a given square, its value is the (arithmetic) mean of the other squares on the board from which a knight can reach the given square in one move. (Knights move in a standard "L"-shape.) For example, using standard algebraic chess notation, the value on the square a1 is equal to the mean of the values on squares b3 and c2.

If the square on the top left of the board (square a8) has a value of 21, what is the value of the square on the bottom right of the board (square h1)?

### August 2021 - The Olympics

Teaser

At the Tokyo Olympics a gold medal weighed 556g, a silver medal weighed 550g and a bronze medal weighed 450g.

What was the total weight of all the medals awarded to athletes during the Olympics?

### July 2021 - The Bag of Marbles

Teaser

A bag contains 100 marbles, and each marble is one of three different colours.

If you were to draw three marbles at random, the probability that you would get one of each colour is exactly 20 percent.

How many marbles of each colour are in the bag?

### May 2021 - An Age Old Question

Teaser

Today, Anna is half the age of Chris.

8 years ago, Anna was half the age of Danielle.

9 years ago, Danielle was half the age of Ben.

When Anna was born, Ben was half as old as Anna is now.

How old will each person be when the sum of all of their ages is 65?

### April 2021 - Dominoes and Chess

Teaser

You have been given 31 dominoes, and a chessboard with 62 squares, where two diagonally opposite corners have been cut away. Each half of a domino is the same size as one of the chessboard squares.

Can you find a way to lay the dominoes on the chessboard in such a way that each square is covered?

### March 2021 - Rock-Paper-Scissors: Sudden Death!

Teaser

You're playing rock paper scissors with your friend. You've played together a lot, and you're quite good at guessing what move they will make: you get it right 50% of the time. You're playing a sudden death match: the first to win one round is the winner, and if you tie, you keep playing until someone wins. 'You can assume that an incorrect guess has an equal chance of resulting in either a draw or a loss'.

What is the probability you win?

### February 2021 - The World's Greatest Game-Show

Teaser

Imagine you are a contestant on a game show. The host has randomly generated a number between 0 and 100 (this could be a whole number or a decimal) and asks you to guess a number lower than it. If you successfully guess a lower number, then you win the amount of your guess. If you guess a higher number, then you leave with nothing.

Assuming you want to maximise your expected earnings, what number should you guess?

### January 2021 - A Game of Pool

Teaser

Three students decide to play a round-robin pool tournament, where the winner stays on and the loser waits their turn to play again.

At the end of the evening, they count the number of games that each of them played:

Ash played 9,

Beth played 13,

Craig played 16.

Who lost the second game?

### November 2020 - Group Stage Analysis

Teaser

A Nations cup group consists of 4 teams who all play each other twice. The teams finishing first and second after all the matches have been played qualify for the knockout stage of the competition.  3 points are awarded for a win and 1 for a draw (0 for a loss). In the case of a points tie a scale of other metrics are used to decide final positions such that no team has the same final position in the group.

What is:

a) the lowest number of points a team can get and still qualify for the knockout stage?

b) the highest number of points a team can get and not qualify for the knockout stage?

A. 4 & 11      B. 5 & 11      C. 6 & 11      D. 4 & 12      E. 5 & 12

### October 2020 - Rock-Paper-Scissors

Teaser

Cari and Morgan play rock-paper-scissors 10 times. You know that Cari uses rock three times, scissors six times, and paper once. Morgan uses rock twice, scissors four times, and paper four times. You know there are no ties in all 10 games but the order of games is unknown.

Who wins and by how many games?

A. Morgan by 6      B. Morgan by 3     C. It is a draw      D. Cari by 2     E. Cari by 4

### September 2020 - The Largest Slice of Pizza

Teaser

100 people are lined up waiting for a slice of pizza they will all be sharing. The first person takes 1% of the pizza. The second then takes person then takes 2% of what is remaining. The third takes 3% of what is remaining and this continues with the nth person always taking n% of the remaining pizza.

Which number in the line will get the largest slice of pizza?

A. 1          B. 3       C. 5         D. 10        E. 25

### August 2020 - A Pressing Partner Problem

Teaser

The APR partners have decided to have new office name plates printed, in order to further increase the respect in which they are held by APR staff and allow them to spend a few hours imagining that they will get back to their offices this year.

The company providing the name plates charges nothing for the plate itself, and instead charges according to the letters being printed. The following quotes have been obtained:

Roger Austin: £19

Gary Heslop: £15

Tim Nash: £11

How much will the company charge if Chris Bryce decides to purchase a name plate (from his own funds, of course!)?

A. £12          B. £14         C. £15          D. £17         E. £0 (Chris should not have a name plate!)

### July 2020 - A Precarious Bridge Crossing

Teaser

Four people are on one side of a weak bridge at night and need to get across. The bridge will only support the weight of two people at a time and a torch is needed to cross safely, but the group only has a single torch. Each person takes a different amount of time to cross - 10 minutes, 5 minutes, 2 minutes and 1 minute respectively - and any pair crossing the bridge must travel at the speed of the slower person.

What is the shortest time in which all four can get from one side of the bridge to the other?

A. Not possible          B. 48 mins          C. 21 mins          D. 19 mins          E. 17 mins

### June 2020 - A Corridor Full of Light Switches

Teaser

100 people walk down a corridor with 100 lights, all initially off. Each person flicks all switches that their number divides. E.g. first person flicks all switches, number 3 flicks 3,6,9,...,99 and the 37th person flicks just 37 and 74.

How many switches are now in the on position after all the people have walked through?

A. 0          B. 10          C. 25          D. 100          E. More information is required

### May 2020 - A Fruit Factory Mix-Up

Teaser

A wholesale fruit factory produces only boxes of apples, boxes of oranges or boxes of mixed apples and oranges. These are labelled by an automated machine however an error has been introduced in the label machines code and now means that every box has been labelled incorrectly.

If you have one of each box which box should you remove a single piece of fruit from to allow you to relabel all boxes correctly.

A. Apples & Oranges         B. Apples          C. Oranges          D. Any of them          E. None of them

### April 2020 - How to Break a Chocolate Bar

Teaser

For a bar of chocolate comprising of 30 squares (a large 6 by 5 bar of your favourite brand say) how many times would you need to break the bar to make it into 30 individual squares?

A. 11         B.15          C. 29          D. 30          E. 31

### March 2020 - A Very Cross Problem

Teaser

Here is an ordinary cross. Drawing only 2 straight lines what is the greatest number of pieces you can divide the cross into?

A. 3          B.4          C. 5          D. 6          E. 8

### February 2020 - A Sequence of Tricky Numbers

Teaser

What is the next number in the sequence?

1, 11, 21, 1211, 111221, …

A. 1122221      B. 1212221      C. 312211           D. 1331212              E. 22

### January 2020 - Mathematicians and Their Names

Teaser

The first six volumes of Encyclopaedia of Mathematics are arranged in order on my shelf from left to right. The six volumes contain mathematicians with surnames beginning; A-Ba, Be-Ca, Ce-Ei, Ek-Fe, Fee-Fi and, Fo-Fum respectively. If one ignores the covers, which of the following encyclopaedia entries could be on the page ‘next to’ the page with the entry for Einstein as they sit on my shelf?

A. Abel       B. Bernoulli         C. Cantor             D. Euler                E. Fibonacci

### December 2019 - An Excel Conundrum

Teaser

Create the shortest possible formula that can be copied from cell A1 downwards to replicate the game "FizzBuzz".

The game works like this:

• The players count up from 1.
• Each multiple of 3 is replaced with the word "Fizz".
• Each multiple of 5 is replaced with the word “Buzz”.
• If a number is both a multiple of 3 and a multiple of 5, then "FizzBuzz" should be used.

Other rules:

• No VBA allowed.
• Your formula has to be written into the cell - no hiding it in a named range or similar.
• That means no row numbers in a separate column, no inputs in fixed cells, no range names, no variables, no formulae in conditional formatting or text formatting, etc.
• Any Excel version allowed, but no add-in or other non-standard functions.
• Array formulae are fine, but the curly braces add 2 to your character count.

### November 2019 - A Game of Tennis

Teaser

A tennis player has a 60% chance of winning any given point in her service game.  Assuming that all points are independent of each other, what is the probability that she will win the service game?

(The scoring system in tennis is such that a player must score a minimum of 4 points and be at least 2 clear points ahead of their opponent in order to win a game.)

### October 2019 - The Prisoners and The Hats

Teaser

One day a jailer brought three prisoners out and had them stand in a line, one behind the other. He then showed them five hats (three red and two white) which he hid in a bag. From these five hats, he selected three and put one on each of the prisoners’ heads. None of the men could see what colour hat he himself wore, but could see the colour of any hats in front of him.

Starting from the back of the line, the jailer asks the first man what colour his hat is, offering him his freedom if he is correct but doubling his sentence if he gets it wrong; he declines to answer. The jailer then asks the second man the colour of his hat, making the same offer; he also declines to answer.

What colour is the third man’s hat?