CCM Puzzle Collection (001-020)

Some mathematical puzzles are delightful. When we discover the principle behind them, they seem so easy ... but they may have had us 'stumped' for hours or days before gaining the insight to solve them!

This is one of those puzzles.

Once I have explained this particular puzzle to you, please pause the video and see if you can work out the solution by yourself. I know the temptation is great to view the answer right away, but you will enjoy the process much more if you 'have a go' first.

You are shown three light switches and told that only one of them controls a light in a distant room (which you cannot observe). You are allowed to manipulate the switches and then you are allowed just one visit to the room to observe the light. You are then required to identify the switch that controls the light!

By the way, when you manipulate the switches, you are not allowed to pull them apart or tamper with the wiring. You are only allowed to turn them off and/or on.

This wonderful puzzle illustrates a mathematical principle very clearly. Please watch the video to learn what it is ... and then try the puzzle on your friends and let me know how you fare.

If you have to organise a knockout competition for 187 tennis players, how many matches must you plan for before you have one winner?

The starting number is not important. You could start with any number you wish. The PRINCIPLE is the important thing! How would you go about solving this problem? I show you the solution and then explain the principle that mathematicians use to solve similar problems. It is quite a clever one ... and stunningly simple!

I challenge you to perform a very simple task ... to cover the squares of a chessboard with dominoes.

But, there is a twist! We first remove two squares from opposite corners. This leaves 62 squares. Theoretically, it should be possible to cover them using 31 dominoes, since each domino covers two adjacent squares. Can you do it?

If you are like most people, you will get or make a board or draw a diagram like the one above and try it out for yourself. After a few attempts, you may begin to believe this is impossible. But ... is it impossible? Is there a clever solution that you overlooked, or is it possible to prove that no solution is possible?

Mathematicians are able to analyse problems like this using a concept of parity ... in this case, the black and white squares of a chessboard. Watch this video and learn how this very clever insight is used to prove whether the problem is solvable, or not!

When you have done that, you might now like to answer these questions: Is it possible to cover the remaining 60 squares if you remove all four corners and only use 30 dominoes? What if you only remove the bottom two corners and use 31 dominoes?

See how simple the puzzles becomes once you have gained this insight? I hope you enjoy the discovery.

The great English puzzler, Henry Ernest Dudeney, had a mischievous sense of humour! In this puzzle, he challenged his readers to start at town A and enter every town just once ... and finish the tour at town Z. You will quickly discover that the problem seems to be insoluble.

How can a mathematician analyse the diagram and demonstrate the possibility (or impossibility) of such a tour? Again, we find that the concept of parity comes to our rescue. By analysing the movements on a chessboard pattern we discover some interesting principles ... and we also discover some of Dundeney's mischievous thinking!

Please watch and enjoy (and learn)!

Mathematicians are not great fans of 'trial and error' or 'guess and check' as a problem solving method. It is true that, sometimes, it is good to explore a problem by 'trial and error' until some pattern becomes evident ... BUT ... mathematicians would always prefer to analyse a problem first and look for some insight or clue that would 'unlock' it.

This classic puzzle simply requires that you place the eight numerals {1,2,3,4,5,6,7,8} into eight squares so that no adjacent/consecutive numbers are placed next to each other. This is a 'packing' problem.

I show that, by examining the most difficult part of the problem/structure, a solution becomes evident.

How many different ways are there of doing something? This is a reasonable question to ask of many activities in life. In this case, we ask how many different ways one can get from one point to another on a grid under certain conditions.

There is a whole class of puzzles dealing with routes from one location to another. Although the setting often varies (e.g. the number of taxi routes from one intersection to another across town), the principles are generally the same.

In this video, I discuss the number of different ways that the word CRYSTAL could be traced out on a 'diamond' grid. I then analyse the number of paths across a chessboard.

This delightful type of puzzle introduces us to Pascal's Triangle, a wonderful mathematical structure. It appears to be so simple in structure, but the mathematics that it opens to us is extraordinary, elegant, and profoundly useful in many fields of mathematics. In time, I intend to produce a series of videos about this triangle but, in the meantime, I encourage you to Google the term (and Google Blaise Pascal as well) and go exploring! Once you understand the principles that I explain in this video, you may wish to print out a map and determine how many different routes there are to get to work or school each day. Your condition may be that you must never take a turn that will take you to an intersection that is further from your destination than where you currently are.

You may find the results surprising! I just discovered that there are 30 different direct routes I could take to the supermarket parking lot just 3.2 km (2 miles) away.

This puzzle has been around for many years. I first encountered it in the 1970s.

If my memory is to be trusted, I think it was in one of Martin Gardner's Scientific American columns, and I believe he first published it in one of his many puzzle books.

The solution is exceedingly simple and requires no complex mathematics, but it eludes many. Be warned! Try it out for yourself first, and then see if your friends can deduce the missing pattern.

Now try this puzzle from the New York Times (April 27, 2009) ...

Sometimes it is fun to watch others struggle with a puzzle that caused you some difficulty! Not everything in life comes easily.

This pattern can be prepared (printed out) and presented on paper ... during a test, for example. It can also be presented by drawing it as people watch ... and I have found this to be the best method.

It shows how easily we can be led astray by details, and how long it can sometimes take before we can see the 'obvious' solution. It is by no means a difficult puzzle. I hope you have fun with it yourself and that you get to try it out on others.

Please watch and enjoy (and learn)!

What Kind of Thinker Are You?

I like this puzzle. No deep mathematical skills are involved, so it is quite ok to present this to children as young as five or six.

My experience is that analytical thinkers (like myself) have more difficulty with it than wholistic or synthetic thinkers. So, this a chance for the non-mathematicians to 'shine!' First, I draw five shapes (or patterns). By the way, it is better drawing them in front of your 'victim' (rather than presenting the complete set at once) because it engages their mind more. Be careful with the way you ask the question. Because ALL the shapes are different, you must ask, "Which of these shapes/patterns is the odd one out?" or "Which one of these shapes/patterns is most different?"

Analytical thinkers focus on what is DIFFERENT about each shape. Synthetic thinkers focus on what is the 'SAME' about each shape.

Please don't jump to conclusions about this puzzle. You will probably be wrong!  The answer is quite counter-intuitive.

If you were standing on the middle rung of a long ladder while it was propped against a wall, what path would your descent follow if the base of the ladder slipped away from the wall?

This is a very practical question, and the answer explains some interesting consequences of such an accident.

Once you understand the problem, pause the video and try to solve it first, BEFORE you watch my solution.

Here's another little challenge that is more difficult than it first appears.

'All' you are asked to do is divide an obtuse-angled triangle into seven acute-angled triangles by drawing straight lines. See if you can do this without looking at the solution. I think you will find that it is deceptively difficult.

I like this puzzle because it requires no 'high-powered mathematics' (no difficult calculations or algebra or complex manipulation of ideas). It requires drawing straight lines and very simple geometry. That is all. Despite this, it allows you to learn some wonderful mathematical principles, some problem-solving logic, and to experience how a mathematician learns by exploring! What do I mean by that? Well, let me ask you some questions ... Can you divide the triangle into a greater number of acute-angled triangles? If so, how, and in what way? Is there a pattern (do the numbers of triangles form a sequence or some such pattern or, beyond a certain number, can you always divide a triangle into a given number of triangles)? What numbers are impossible to achieve?

Apart from dividing obtuse-angled triangles I can challenge you with a particularly interesting problem ... can you use the solution to this puzzle to divide a square into TEN acute-angled triangles? What other geometrical knowledge (intuitive or formal) would you need to use in order to achieve this goal? This will be the subject of my next video, so please watch it, too.

Finally, if you can divide a square into ten acute-angled triangles, can you divide it into fewer acute-angled triangles? Can you divide it into five or six such triangles? What about other numbers? What about other quadrilaterals? I encourage you to go exploring and see what you can uncover. This is how a mathematician learns.

This video is a sequel to my video (immediately above) about dividing an obtuse-angled triangle into seven acute-angled triangles.

In that video I challenged you to divide a square into ten acute-angled triangles. Instead of repeating the challenge here, I decided to give the solution instead ... and discuss an interesting geometrical theorem/property that is needed in order to fully solve the problem. The mathematics of dividing shapes with straight lines is very simple in concept but can lead to extraordinarily complex mathematics.

In short, geometry (playing with shapes) can provide simple enjoyment for anone willing to draw, fold or cut shapes, but can occupy the best of mathematical minds for a lifetime. There is always something to discover ... just keep asking questions. After you have seen this video, see if you can divide a square into fewer acute-angled triangles? Can you divide it into five or six such triangles? Is ten the best you can manage? What about other numbers? What about other quadrilaterals? I encourage you to go exploring and see what you can uncover. This is how a mathematician learns.

Let me introduce you to Tartaglia, an Italian mathematician from the early 1500s. This man had a very difficult upbringing. During his childhood, his dad was murdered, and while he was sheltering during a war a soldier slashed at him, severely damaging his lower jaw and palate so he could never speak properly again. He was called Tartaglia ('stammerer') and was basically self-taught, having almost no access to formal schooling.

He became one of Italy's greatest mathematicians, writing the world's first treatise on ballistics and pioneering a number of solutions to cubic equations. Let me recommend that you Google his name and learn about him.

He is also famous in mathematical circles for a particular kind of strategy problem ... a very practical one in his day. It involved measuring a specific volume of water from a tank/reservoir using two unmarked jugs. In his puzzle, you would be given a 3 unit and a 5 unit jug (let's call them litres) and asked to measure out 4 litres. Many such problems now exist. By the time this video is finished, you should be able to work out your own challenges.

Since his day, mathematicians have developed diagrams/patterns that allow them to find out winning strategies quickly. I explain how to draw and use these diagrams in the video.

Let me encourage you to experiment and explore ... to find out if there are certain combinations that will NOT work, and why. Perhaps you can work out some principles involved and, maybe a formula that will predict the number of moves that you will need for the simplest/quickest solution.

Don't be satisfied with my explanation. I have not told you everything. Could you measure 4 litres with a 6 and 8, a 9 and 3, a 5 and 8 ... etc.? You will be amazed at the things you will discover if you explore this puzzle diligently.