# Go Figure the best problem solving strategy

In a break with tradition I am going to choose a puzzle of the month that I have already used as the monthly puzzle a couple of years ago. The reason is that two weeks ago I heard the most wonderful new solution to the puzzle that I’m sure you will appreciate so let’s start with the puzzle:

Three people enjoyed a meal at a restaurant. The waiter brings the bill for £30 so each person pays £10. Later the chef realises that the bill should have only been £25 so he sends the waiter back to the table with five pound coins. The waiter could not figure out how to divide the £5 so he gave each person a £1 and kept £2 for himself.

So….the three people have paid nine pounds each for the meal:  3 x £9 = £27
The waiter kept two pounds:   £27 + £2 = £29
What happened to the other pound?

The new answer will be at the bottom of this newsletter but before that here are some of the new resources added to the website this last month.

Go Figure is a number placing puzzle in which interconnected addition, subtraction. multiplication and division calculations have to be completed using the digits one to nine.

I got quite excited when I saw pupils using this activity for the first time and heard them talk about their insights. The puzzle can be used to introduce a new problem-solving strategy for this kind of task. Rather than concentrate on which digits could go in the available spaces, make a list of the digits that could not possibly go into the spaces. You really need to try this yourself to see how the properties of the four rules are analysed in the puzzle solving thinking. Make sure you click the ‘Show Tags’ button to assist you find the solution.

Olympic Rings was put together to coincide with the Winter Olympics but the relevance of the puzzle will live on during this inter-Olympic time. There are three levels of difficulty with the lower levels being made easier with some pre-placed digits. This makes the puzzle accessible to younger children but also provides a starting point for an advanced level proof investigation.

Map Scales came about because, after being asked for a good exercise on ratios as used in map scales, I couldn’t find one! There are two levels and the second level introduces the tricky and not necessarily intuitive notion of area scale factors.

Barmy BIDMAS Is a new advanced Lesson Starter. You will need to know about the order of operations and factorial notation to appreciate the subtly of this mathematical wonder. Students could be challenged to make a similar calculation with the surprising value of 6!

Time Sort is the latest additions to the ‘Telling the Time’ collection. There are three levels including digital times, analogue clocks and phrases to represent times. Try using it with pupils working in pairs and listen to the discussion generated.

Sum to One is a set of decimal numbers on virtual cards which can be used for a matching activity. A pairs game, a multiple choice quiz, a tug-of-war game and a snap game. Is that too much choice?

The book I am reading at the moment is Craig Barton’s new book How I Wish I’d Taught Maths: Lessons learned from research, conversations with experts, and 12 years of mistakes. I am a great fan of Craig’s podcasts (I listen to them on my Tuesday morning cycle ride) and this book collects together the insights Craig has collected from all of the educational experts he has interviewed. At the time of writing, 93% of the reviewers on Amazon had awarded the book five out of five stars. I thoroughly recommend this book to you here.

Thanks to those of you who have posted photographs on Twitter of the Transum activities being used in the classroom. It is so good to see that the work that went in to producing the resources was worthwhile. Thanks

If you follow me on Twitter (@Transum) you may have noticed that my list of ‘Hidden Gems on the Transum Website’ has been growing from the 19 included in the last newsletter. I think I will stop when I get to 50.

Now let’s continue the search for the missing pound from the puzzle of the month. One hour later two elderly ladies came into the restaurant and enjoyed a meal together. The waiter brings the bill for £30 so each lady pays £15. The chef again tells the waiter that the bill should have only been £25 so he sends the waiter back to the table with five pound coins. The waiter could not figure out how to divide the £5 so he gave each lady a £1 and kept £3 for himself.

So….the two ladies have paid fourteen pounds each for the meal:  2 x £14 = £28

The waiter kept three pounds:   £28 + £3 = £31

So there is the missing pound! Genius isn’t it? I heard this solution on the Danny Baker radio show and have included the audio excerpt towards the bottom of the June 19th Starter of the Day. It’s worth listening to. Enjoy.

All the best for the month ahead

John

P.S. Always wear glasses to Maths lessons. They help with division!

# A Happy MMXVIII From Transum

Happy New Year. I hope that 2018 proves to be a good, positive number for you and that you, and your pupils, achieve all that you want during the next twelve months. If the Roman numeral in the title caught your eye you may like the Roman Numerals Quiz.

This month’s puzzle is taken from the excellent book I have been reading during the holiday called “Can You Solve My Problems” by Alex Bellos. I have just read the problem called “The Shrivelled Spuds” which I present to you here:

A pile of potatoes weighing 100kg is put in the sun. Ninety nine per cent of the weight of the potatoes is made up of water. After a day some of the water evaporates., with the result that 98 per cent of the weight of the potatoes is now made up of water. What’s the new weight of the potatoes?

I thoroughly recommend the book as not only is it an ordered collection of intriguing puzzles but it also has an extensive solution section in which Alex provides insights, history and worked solutions for the puzzles. The chapters are Logic Problems, Geometry Problems, Practical Problems, Problems with Props and Problems for Purists. Here is a link to buy the book from Amazon.

Last month the website was added to and updated as usual but the one new activity I would like draw your attention to is the Area Wall Puzzles. The core concept is a puzzle called Shikaku, an original Nikoli puzzle and though the Transum version refers to area, the activity requires the ability to consider factor pairs of small numbers.

In the process of creating and testing the puzzles I realised how addictive this type of problem solving is. Just like Sudoku solving you will develop strategies as you become more proficient and experience a nice sense of accomplishment when the wall is completely coloured in.

Another new activity is called “Equation of a Line Through Points“. It is a four level exercise requiring users to match the equations of the straight line graphs to the clues about gradients and points. This exercise could be attempted after some of the more basic “y=mx+c” activities have been mastered.

Finally, my answer to this month’s puzzle is 50kg

I found this by first realising that if 99% of the original pile is water then 1% must be other, dry matter.

1% of 100kg is 1kg.

Let the weight of the potatoes after the drying be x.

0.98x + 1 = x

1 = 0.02x

x= 50

That’s all for now

John

P.S. You have to be odd if you want to be Number One.

# October 2016 News

This is the Transum Newsletter for October 2016, the 10th month of the year. Have you ever noticed that the month name begins with the suffix ‘Oct-‘ suggesting eight and not ten. There is a reason for that and a quick internet search will reveal it to you.

Let’s begin with the puzzle for this month which is about three hungry children.

There was a short queue in the school canteen. Ayden was directly in front of Betsy who was directly in front of Carl.

Aden’s age is an even number but Carl’s is odd. Is a person with an even age directly in front of a person with an odd age? The answer is at the end of this newsletter.

I am very keen to tell you about some of the new additions to the Transum website that appeared last month. The first is Maths Mind Reader. Absolutely everyone I’ve used it with have been extremely impressed with this clever web page. As a Transum subscriber you will be see the mathematics that makes it work and Secondary pupils should be able to understand and even prove the concept.

A Transum website visitor, Les Page, sent me an addictive little puzzle he has devised called Zygo. He has kindly allowed a Transum interactive version to be created which is now ready to improve the numeracy and problem solving skills of your pupils. Thanks Les.

Pupils quickly learn to recognise and name regular polygons but the new activity called Polygon People may help younger pupils to name irregular polygons too. The activity has three levels and only accepts the correct spellings.

For the older pupils (14+) the Completing the Square and Proof of Circle Theorems activities should support those entered for the higher tier of the GCSE exams (or equivalent).

At times when I have not been creating new content for the website I have had a small amount of time to look at an updated app that I have downloaded to my iPhone. Photomath has been around for a couple of years but I’ve been very impressed with the recent improvements. You point your phone camera at an equation, and it will give you the answer and show you the working. I’m still amazed it can read my handwriting!

Photomath supports arithmetic, integers, fractions, decimal numbers, roots, algebraic expressions, linear equations and inequalities, quadratic equations and inequalities, absolute equations and inequalities, systems of equations, logarithms, trigonometry, exponential and logarithmic functions, derivatives and integrals.

My only reservation against using it with pupils is some of the phrases used to explain the stages of solving an equation. “Move constant to the right side and change its sign. Move variable to the left side and change its sign” is less helpful than the notion of doing the same thing to both sides in my opinion.

The answer to this month’s puzzle is yes. We don’t know Betsy’s age but we do know it is either even or odd. Let’s consider the two possibilities.

If Betsy’s age is odd then Ayden (even) is in front of Betsy (odd) and the answer is yes.

If Betsy’s age is even then Betsy (even) is in front of Carl (odd) and the answer is yes.

So regardless of Betsy’s age, the answer is always yes.

A similar problem was devised by Hector Levesque and it was included in Alex Bellos’ Guardian blog. Unbelievably 72 per cent of the 200,000 people who answered the question got it wrong.

That’s all for this month.

John

P.S. Why do mathematicians think that Halloween and Christmas are the same?

Because 31 OCT = 25 DEC (You need to know about the octal number system to understand this month’s joke 318 = 2510)

# December 2015 News

Happy Christmas. This is the December Transum newsletter coming to you with lots of festive cheer and good wishes for the future.

Here is the puzzle for this month. Noel and Merrie received some Christmas presents. The number of presents Noel received was a power of 3. The number of presents Merrie received was a power of 2. The number of Christmas presents they received were consecutive numbers and less than ten. How many presents did they receive?

You will probably arrive at a first answer fairly quickly but there are actually four different answers. Can you find all four? The answers will be given at the end of this newsletter after a rundown of the notable new features on the Transum website.

Although they are not a new idea, many teachers have never used mini whiteboards in class. The fact is that most classrooms are more likely to have pupils with computing devices than mini whiteboards so this simple facility allows you to make alternative use of the devices. Pupils write or draw their answer to plenary questions on this whiteboard simulator then hold up their devices so the teacher can see. This is a much better idea than putting hands up! This way the teacher can instantly see what everyone is thinking rather than just one person.

I would like to say that I am surprised that the Broken Chessboard Puzzle has been completed by so many people. The puzzle was first made popular by Henry Ernest Dudeney (1857 – 1930) who was an English author and mathematician. The puzzle appeared in his book ‘The Canterbury Puzzles and Other Curious Problems’ in 1907. I thought solving the puzzle would be too difficult for most people but I have been proved wrong. Congratulations to all those people who have earned a trophy for finding a correct solution.

The Starters for 6th March and 4th December have both been replaced with new ideas. The first is the Goat Grazing situation I have used for decades to introduce the topic of Loci. The latter challenges pupils to imagine a dice reflected in two mirrors. Quite a challenge for many.

The number of printable worksheets available only to subscribers is rapidly increasing. Even in these days of hi-tech it a good idea to provide variety to the learning experience with more practical activities particularly relevant to some Maths topics more than others.

Finally I would like to say how much I have enjoyed a new app I downloaded (free) to my iPad. It is called Sumaze and has been produced by MEI. Perfect for Year 12 students who need practice with inequalities, the modulus function, indices, logarithms and primes. It is also useful lower down the school but certain sections such as the logarithm puzzles would of course be inaccessible. Solving the thoughtfully constructed puzzles brings an enhanced understanding of the basic mathematical concepts in a fun setting. I’m stuck on the final level in Fermat’s Room so any suggestions gratefully received.

Are you having a problem thinking what to write in the Christmas cards you are sending to your Maths teacher colleagues? How about this:

It reads ‘Complements of the (Cs) season (2u) to you’.

Talking of Christmas and puzzles, here are the four answers to this month’s seasonal puzzle which is also the Starter for 16th December and called The Power of Christmas:

Have a happy and merry Christmas and enjoy the Transum festive activities.

John

ps Q. Why did the pupil get closer to the fireplace when doing Maths homework?

# October 2015 News

In the days before Wikipedia and Google we might refer to an encyclopedia to find answers to our questions. The puzzle for this newsletter is based on a set of ten volumes of an encyclopedia on a bookshelf arranged in order with volume one on the left and volume ten on the right.

A bookworm eats through from the front cover of volume one to the back cover of volume ten. What is the length of the bookworm’s meal if each encyclopedia is 5cm thick (the pages are 4cm and each cover is 0.5 cm thick)?

The answer is at the end of this newsletter but, be warned, it is not the obvious answer.

This puzzle is a version of the February 2nd Starter of the Day which presents a random number of encyclopedias and randomly generated measurements for the pages and covers. It provides an opportunity for pupils to engage in some decimal addition and multiplication before being surprised by the actual answer.

Now the pages on the Transum website should be a little easier to find as the search facility has been upgraded. Now when you search for a term you get two sets of results. The first is directly from the Transum database and is a search on page titles and descriptions. Lower down the page you will see the Google search results which include snippets of text found on the pages.

You may like to try out the new search feature to find this month’s new additions. Firstly the Car Park Puzzle challenges you to get your car out of the very crowded car park by moving other cars forwards or backwards. It is the Transum version of a puzzle that has been available in different formats for many years but the real challenge for students is to devise a level 6 puzzle that is possible but requires more moves than level 5. The way the students record moves and consider the advantages of working backwards (doing and undoing) give this challenge a strong mathematical connection.

Polybragging is another new activity that is also based on an idea that has been around for a long time. This is a game for two or more players. Each played needs a tablet, computer or smartphone with the page loaded.

If you have ever played a card game called Top Trumps you will know the main idea of this game already. Each player is given a shape that the computer selects at random. The players each choose a shape property and whichever player has the highest value for that property wins a point.

The properties available include the size of the largest interior angle, number of pairs of parallel lines, number of lines of symmetry and the order of rotational symmetry.

Hopefully, by playing the game, pupils will develop more familiarity and a greater knowledge of the properties of polygons.

Other new additions to the site include a Dice and Spinners page to use if you can’t find the real things and a Reaction Time activity which collects data about how quickly we recognise even compared to odd numbers.

Finally some more traditional Maths exercises have been added. These include Multi-step Problems and Decimal Times. These exercises are self-marking, printable and every pupil gets a slightly different version thanks to the in-built random number generator.

Thanks to everyone who has added comments and suggestions to the site this month. Your input keeps the site alive. One comment waiting for your opinion is that made by Leslie Jackson on the 16th December Starter page. Do you think powers of two are 2, 4, 8 .. or do you think they are 4, 9, 16 …?

Finally the answer to this month’s puzzle. The answer is not 50cm surprisingly. If you picture the ten volumes arranged on the shelf you will notice that the front cover of volume one is actually on the right, next to volume two! So if the bookworm starts by eating through that cover it has missed the pages and back cover of volume one altogether. Similarly the back cover of volume ten is on the left so the bookworm stops before eating the pages and front cover of volume ten.

The correct answer is 50cm – 2(4cm + 0.5cm) = 41cm

Enjoy October and don’t miss the Halloween Starter at the very end of the month.

John

ps What do you get if you divide the circumference of a Halloween lantern by its diameter?

A: Pumpkin Pi

# April 2015 News

Welcome to the April 2015 edition of the Transum Mathematics Newsletter. Did you start the month off with the April Fool’s Starter? Did your pupils fall for it?

Your puzzle for this month is about a game called Best Dice in which two people roll a dice and whoever gets the higher number wins. A prize is awarded to the person winning most times after 100 games. The catch is the dice don’t have the numbers one to six on their faces.

There are four different dice and you are allowed to choose which dice you will play with.

Best Dice

• The red dice has threes on all of its faces.
• The blue dice has four fours and two zeros.
• The yellow dice has three fives and three ones.
• The green dice has four twos and two sevens.

Which dice would you choose to give you the best chance of winning the prize? The answer can be found at the end of this newsletter.

There have been many pages added and updated during this last month. A new puzzle called Numskull  is designed to provide a relaxing logic challenge where the mathematics involved is suitable for upper Primary pupils. There are five levels differing by the number of clues available.

For older students a Number Systems Venn Diagrams activity provides a quick but effective revision task. The objective is to drag the numbers in to the correct layer of the concentric circles. The software checks the correctness of the placings.

Also for older students is a rapidly growing database of Exam-Type Questions and their worked solutions. There are currently 90 questions and answers in the database but more are being added regularly. They are similar to questions that have appeared on IB Standard, Maths Studies and GCSE examinations but have all had the wording and numbers changed to make them different to past-paper questions you may find elsewhere. The solutions can be revealed line by line making a great teaching tool for the classroom.

Though not specifically mathematical a Scheduling puzzle has been added to provide a little more variety to the Transum Puzzles page. It’s not too difficult and the software shows you which criteria you have and have not fulfilled when you choose to check your solution. I’d love to know if you decide to use it with your pupils.

I’m not sure how we managed so long with out a traditional fractions, decimals and percentages  conversion activity on the Transum website. Now there’s a Starter, an interactive pupil activity and a revision presentation on this important topic.

The answer to the puzzle posed at the beginning of this newsletter is a bit like rock, paper, scissors. Whichever dice you choose, your opponent could always pick one of the remaining dice which has a better chance of beating you in the long term. Construct the possibility spaces for the possible dice pairings to see for yourself.

Blue beats red, red beats green, green beats yellow and yellow beats blue! You can see why in the answers section of the Best Dice Starter page.

Have a happy Easter, Songkran or whatever you may be celebrating in April.

John

ps . What do you call a saucepan of simmering soup on top of a mountain?

… A high-pot-in-use!