Discover Edinburgh's Mathematical History

Update for 2022: some of the locations now include audio descriptions written and recorded by students and staff at the University of Edinburgh.

Welcome to your tour! Below you can find:

• A map with three suggested routes to explore Edinburgh's mathematical history (you should also feel free to visit any of the locations in any order).
• An interactive map with interesting historical and mathematical facts about famous, and less famous, Edinburgh locations. Some also contain audio descriptions.
• A mathematical puzzle for each location (these can all be found below the maps). We encourage you to take advantage of Edinburgh's lovely cafés, pubs, and outdoor locations to stop on your walk and try some of these fun puzzles.
• For ease of navigation, we also recommend opening a static version of the map, with written directions, which is available on the  Maths Outreach Website .
• The Green route will take approximately 1/2 hour (1 mile/1.5km), the Blue route takes 2.5-3.5 hours depending on stops/reading; it can be walked in about 1.5 hours (3.5 miles/6km), and the Red route will take an extra hour (2.5 miles/4km extra).

Below you will find a fun puzzle (or two) with links to each of the locations. If you get stuck, or would like to check your solutions, the solutions and explanations are available on the  Maths Outreach website .

Witches' Well Puzzle

It is the 16th century in Scotland, and two villagers are accused of being witches.  

The general says that there is a very simple test to decide whether they are witches. Each of the two witches picks a card and can look at it if they like. They are then put in separate rooms, cutting off any form of communication between the two witches. Their task is to try to predict the colour of the other witch’s card.  

If they are both wrong, or one of them is wrong, they are free to go. Otherwise, if they both correctly predict the colour of the other witch’s card, then they are assumed to be in league with the devil and will be burned at the stake. The cards can either be red or black 

Before choosing their cards, the two women have 1 minute to come up with a strategy. What is their best strategy to survive? 

Camera Obscura Puzzle

A pinhole camera is an example of the  camera obscura , which is Latin for a vaulted room (camera) which is dark (obscura).  A camera obscura projects an image of its surroundings onto a screen, and is an early fore-runner of modern photography. You can make one yourself with a cardboard box and some photographic paper. The pinhole camera (and camera obscura) works by reflecting light off an object which is then focused onto a surface, creating an image of the object. In a modern camera, focusing happens through the use of a lens, but in a pinhole camera it is the pinhole itself which focuses the rays of light.  

Here we look at a puzzle concerning how large a pinhole camera would have to be to get a nice picture of the Scott Monument (which you may also see on this walk), which is in Princes Street Gardens and stands 60m tall. If the monument is located at a perpendicular distance of 80m from the pinhole, how far away from the wall in the darkened room must the pinhole be for the projected image to be 1m in height? 

As a hint, you may know that two triangles are similar if their angles are the same and their corresponding sides are in the same ratio. Since the wall in the darkened room and the Scott monument are arranged in a parallel manner, the two triangles shown in the diagram above are similar.

Peter Guthrie Tait Puzzle

Here is a challenge for you; will need to use a scarf (if you are particularly lucky and don’t need a scarf today, you can also use any long rope or piece of string...).

Challenge: Pick the two ends of the scarf with your hands, and tie a knot without ever letting go of the ends. 

James Clerk Maxwell Statue Puzzle

Maxwell was involved with developing colour photography by combining 3 black and white photographs taken with red, green, and blue filters in front of the lens. Can you work out which of the black and white images below corresponds to the red, green, and blue parts of the picture of two macaws? You’ll need to know that white parts mean lots of that colour, and black parts little of it. You may also need to look into which colours of light mix to make others, for example, how is yellow made from red, green, and blue light? 

Scott Monument Puzzle

Walter Scott’s father was a Freemason, and after his father’s death in 1799, Scott himself joined the order. The Freemasons are known to have frequently used a simple  pigpen cipher  to encrypt private messages and information – in fact, they used it so often that the pigpen cipher is sometimes called the Freemason cipher. Each letter of the alphabet is replaced by a symbol, shown by the picture below. 

As an example, we decode the sentence below: 

Of course, it is unknown whether Scott himself would ever have used such a system; but try your hand at unravelling the following sentence? How is it related to Scott? 

City Observatory Puzzle

The city observatory houses two telescopes: A Fraunhofer-Repsold Transit Telescope, and a 6-inch astronomical observatory refracting telescope, made by T Cooke. In 1839, the first Scottish Royal Astronomer,  Thomas Henderson  was one of the first people to measure the distance to a star. He managed to measure the distance to  Alpha Centauri , which we now know is the closest star to the sun, with reasonable accuracy.  

This puzzle will help you to use mathematics to work out how long light takes to reach us from Alpha Centauri.  The accurate distance of Alpha Centuri is now known to be 4.1315 x 10^16 meters from the Earth. If we were to look at an image of Alpha Centuri through a telescope today, how old would that image be? 

Hint: You may need to look up the  speed at which light travels . 

Holyrood Abbey Puzzle

Imagine that we are in the 16th century and you know where to hide from your potential debt collectors. May I interest you in a game?

Game 1: (The St. Petersburg paradox) 

For this game you will only need a coin. For a small fee, I will let you play a game where you never lose, but it may take you some time to win. At the start of the game, we have £2 in the pot. For each round, we toss the coin and there are two possible outcomes:  

If you win in the 1st round, you win £2. If you win on the 2nd round you win £4. If you only win on the k-th round, you win £2^k.  

1. How many rounds do you need to lose before you can win a million pounds? 
2. How likely is it to lose those that many times in a row? 
3. How much would you be willing to pay to play this game?

Game 2: (The Martingale betting system) 

For this game you will need a six-sided die (or you can  roll one online ). We both bet the same amount X and roll the die. If the number is less or equal to 4, you win the pot (2X), otherwise I win. 

What if I told you that there is a strategy that guarantees that you always make a profit? 

If you win the round, great! If you lose, you double your bet (and I put in the same). This is known as the Martingale betting system. 

Try playing the game with a starting bet X (=1 or 2). How long did it take to win? How much profit did you make?  

Scottish Parliament Puzzle

There is lots of interesting mathematics in voting. For example,  Arrow’s impossibility theorem  demonstrates that, where there are three or more options to vote for, there is no ‘fair’ voting system. This puzzle looks at a different aspect of voting:  gerrymandering , which is when someone manipulates voting district boundaries to the advantage one a particular group.   

Imagine that your job is to draw the boundaries for voting regions, and you happen to be a member of the Pink party (who are competing against the Grey party). There are 25 voters (squares) in the picture below, which you must divide up into five regions, each of which must contain five squares. No regions can overlap, and each square in a region must share a side with at least one other square in that region. Whichever party has more votes in a region wins. Notice that there are 9 Pink voters and 16 Grey voters, so the Grey voters are winning overall. Can you form five regions so that the Pink party wins three of them?

The World's End Puzzle

Here are two puzzles ideal for trying while you're sitting comfortably in a pub or café:

Puzzle 1: (Finding pi)

While you’re having a sit down, why not try to estimate the value of pi? We’ll use an experiment called  Buffon’s needle . We’re going to be using a piece of paper, a pen/pencil, a ruler, and some matches, sticks, needles, or other long, thin objects. If you don't have these things with you then you can  try it online . First draw some parallel lines on your paper which are the same distance apart as the length of your ‘needles’. Put your piece of paper on a table and drop the ‘needle’ onto it. We’re going to do this many times and keep track of whether the ‘needle’ crosses a line or not. Keep two tallies of these cases while you keep dropping ‘needles’ (you can use more than one, as long as they are the same length). For example, in the picture below there are three needles (in red) which do not cross a line, and two (in green) that do. 

How is this related to pi? Well, if you compute 2x(number of drops)/(number of hits), this should approximate pi. You may need to do quite a few drops, or use the online versions below. You may want to keep track of the computed value as you drop more and more needles. How close to the actual value of pi (which is approximately 3.14159265359) do you get? 

Puzzle 2: (Surprising sizes)

While you are having a drink, have a look at the glass you are drinking from, and try and guess the answer to this question: 

What is longer, the circumference of the glass (i e the edge) or the height? 

Now try and measure both the circumference and the height (try and do it without spilling your drink). 

Was your guess correct? 

Try with different kinds of glasses...  

What did you find? Did your findings surprise you?

The Oyster Club Puzzle

Here is a puzzle about doors.

Suppose you are on a game show, and you have won a chance at the grand prize. On this game show you choose one door of three. One door hides the grand prize, and the other two have a booby prize behind them. You choose a door but do not yet open it. The host then opens a door that you have not chosen, revealing that it has a booby prize behind it. They then offer the choice to switch your choice to the other as yet unopened door.  

What strategy maximises your chance of winning the grand prize? What are the chances of winning given that you do not switch, and what are the chances of winning if you do? 

Greyfriars Kirkyard Puzzle

Another mathematician buried at Greyfriar’s is  William Wallace . He is one of three mathematicians credited with proving the Wallace-Bolyai-Gerwien Theorem: any polygon can be cut into pieces and rearranged to match any other polygon with the same area. 

By cutting the following triangle, see if you can rearrange the pieces to form a square.

Another application of this theorem is the famous ancient Chinese puzzle tangram. You can try out some  tangram challenges online here .

Harry Potter Puzzle

Arithmancy has charts which can be used to gain a numerical interpretation of a person’s name. By identifying numbers associated with specific letters in a name we can calculate the ‘Character Number‘, ‘Heart Number’ and ‘Social Number’, describing an individual’s general personality, their hopes and desires and outer personality. 

Below you can see an arithmancy chart, where each letter in the alphabet is assigned a number. Using this chart we can convert any name into a number. 

For example: HARRY = 81997 

From this, we need to sum the digits together repeatedly until a single number is reached. This number is called the ‘digital root’. 

 For example: 8+1+9+9+7=34 

                         3+4=7 

So, 7 is the digital root. 

The Character Number is obtained by using all the letters in the name, the Heart Number using all the vowels and the Social Number using all the consonants. 

Minerva McGonagall, a teacher in Hogwarts and the most notable Scottish character in Harry Potter. Use the chart to calculate her Character, Heart, and Social Numbers.

Royal Mile Puzzle

The Crown of Scotland, the Sword of State, and the Sceptre together form the  Honours of Scotland  - essentially, the Scottish equivalent of the Crown Jewels. From the Restoration of Charles II in 1660 to the Treaty of Union 1707, the state opening of Parliament was marked with great ceremony by bringing the Honours down from Edinburgh Castle, and taking them to Parliament House on the Royal Mile adjacent to St Giles Cathedral. A stream of attendant Lords and Barons and other nobilities, all dressed in finery of many colours, would accompany the Honours on their journey. This elaborate procession from Edinburgh Castle down the Royal Mile to Parliament became known as the  Riding of Parliament . 

Here's a brainteaser. Suppose the year is 1685, and the Riding of Parliament has just begun. One of the many Lords has a message he wishes to send urgently to one of his contacts awaiting him at Parliament House. With so many people involved, the Riding of Parliament must proceed at a snail's pace of half-a-mile per hour, so he sends his manservant to deliver the message and report back immediately. Hurrying along, his manservant can keep a pace of five miles per hour. He gets to Parliament, does his job, immediately turns around and returns to his Lordship, who by now has travelled a little way down the Mile. The Lord then sends him back to Parliament with another message, and the manservant goes back and forth like this until the Riding of Parliament finally ends. 

Supposing that the distance from Edinburgh Castle to Parliament House really was just one mile, how far would the manservant have run in total? 

Luckenbooths Puzzle

Below are three images of a circle of radius 1.5cm on grids of different scales. We can use the grid squares to estimate the area of the circle. To do this we multiply the area of the squares by the number of those squares whose area lies mostly inside the circle. 

In the first image the squares are 1cm^2, in the second the squares are 0.25cm^2 and in the final image the squares are 0.04cm^2.  

Estimate the area of the circle using the 3 different grids.  

Now calculate the exact area of the circle using the formula A=pi*r^2.

How does the exact answer compare with your estimations? 

Do you notice anything happening to your estimates as we make the grid squares smaller?

Surgeons' Hall Puzzle

A dentist plans to carry out a wisdom tooth extraction on a patient and needs to apply local anaesthetic in the mouth to relieve the pain. However, there is some information about the patient missing, namely their weight, which is required to calculate the correct volume of anaesthetic to administer.  

It is known that the patient has  BMI  of 22.1 and a height of 5 ft 7 inches. The anaesthetic that will be used during the procedure has a concentration of 0.5% and a maximum allowable dose of 7 mg/kg. 

Can you use the following formulae to calculate the maximum allowable volume in mL that should be used to anaesthetise this patient?  

• BMI = weight (kg)/ [height (m)]^2 .
• Maximum allowable volume (mL) = maximum allowable dose (mg/kg) x (weight in kg/10) x (1/concentration of local anaesthetic in percent)  .

Bayes Centre Puzzle

 Bayes' Theorem  is stated mathematically as the following equation: 

P(A|B)=P(B|A)P(A)/P(B) 

Where, 

• P(A) is the overall probability that the event A occurs 
• P(B) is the overall probability that the event B occurs 
• P(B|A) is the probability that the event B occurs given that A has occurred 
• P(A|B) is the probability that the event A occurs given that B has occurred 

The formula is used to calculate the probability of an event A happening given the occurrence of an event B.  

Bayes formula is commonly applied to problems associated with medical testing, for example, in checking how accurate a test is at detecting a disease.  

The table below shows the probabilities associated with testing for a rare disease.

We can read the table like so,  

• The probability of having the disease is 0.01  
• Given that the patient has the disease, the probability of testing positive is 0.8  
• Given that the patient has the disease, the probability of testing negative is 0.2

Now,  let

• A = the event that the patient is disease free 
• B = the event that the test result is positive 

The probability of receiving a positive test result is 0.10304. Given that a patient has received a positive test, what is the probability that the patient is not suffering from the disease? (This is known as the probability of a false positive test result!) 

St Albert's Chapel Puzzle

A group of explorers walk one mile South, then one mile West, then one mile North. At this point, they find out they are back where they started! What was their starting point on Earth?

The Meadows Puzzle

The origins of graph theory can be traced back to a paper published in 1736 by the Swiss mathematician  Leonhard Euler . This paper dealt with a puzzle regarding the seven bridges in the city of Königsberg in Prussia (now Kaliningrad in Russia), in particular, whether it was possible to design a walk that would cross each of the  bridges  once and only once. 

Below you will find a map of the western part of the Meadows. Starting anywhere you like on the map, can you go through each path once and only once? 

(Are you stuck? Try counting the number of paths leaving each point and see if you spot the two special ones.) 

What if you remove the path which is crossed out in red below? (Let's pretend that it's currently closed for repairs.)

Canal Basin Puzzle

The canal basin forms one end of the Union Canal – the other is near Falkirk where it joins the Forth and Clyde Canal. In this puzzle we’ll be looking at how long it would take a soliton wave to travel the length of the canal. First, we need to know how long the canal is. You can work this out by measuring it on the picture below – the red line is 1km long, which gives you a scale to work from. You can see an interactive version at  open street map , and another version in the canal basin.

Now we need to know how fast a soliton wave travels. Russell (see the Canal Basin entry for more information on him) discovered that the speed of a soliton depends only on the acceleration due to gravity, g = 9.81 m/s^2, and the depth of the canal, d, which we will assume is 2m. The formula is

speed = (d g)^(½). 

Once you have the distance and the speed, you can answer the following questions: 

1) How long would it take for a soliton to travel the length of the canal? 

2) Do you think this is faster or slower than you could walk that distance? (It may help to convert the speed into km/h or miles/h.) 

3) What about if you were riding a bike? 

Thank you for taking part in this tour. This is the first year that we've created these self-guided tours and we'd love to know how you found it. We would be grateful if you could spend a couple of minutes completing a short survey about your experience.