# 9 Trafalgar Square Puzzles – One unsolved

You have just begun reading the Transum Newsletter for September 2018 and, as usual, it begins with a puzzle for the month.

My clock does not have any numbers on its face, just markers for each hour/five-minute interval. I looked at it in a mirror one morning and noted the time it appeared to be showing. An hour and a half later while eating breakfast in the kitchen I noticed the clock on my phone is showing the time the reflected clock appeared to show earlier.

Assuming that both clock and phone were showing the accurate time, what time was it when I first viewed my clock in the mirror? The answer is at the end of this newsletter.

The majority of Transum subscribers live in the northern hemisphere so a Back To School theme is appropriate. There are many ideas and resources on the Transum page created for this time of year. Please let me know if you have any other suggestions for teachers meeting classes for the first time.

I stayed in London for a while during the summer and was delighted to see that a pavement artist in Trafalgar Square had drawn a set of maths puzzles instead of the usual art seen in the area. I photographed the puzzles and created an interactive version which are now online. Each puzzle is in the form of a three by three square containing numbers linked by hidden rules. I have named this kind of puzzle as a Trafalgar Square! (Can you see what I did there?)

Thanks to help from some clever people who have seen my photographs online already I have figured out the answers to all but one of the puzzles drawn in chalk by the artist from Slovakia. If you can help solve the puzzle marked Level 8 I will be eternally grateful.

Another new addition to the website is called Vector Cops. Teachers of an older persuasion may recognise the idea from a program popular in schools in the 1980s called Vectmeet, originally published by SMILE (Secondary Mathematics Individualised Learning Experiment). I have created ten levels of difficulty hoping to achieve a low threshold and a high ceiling.

I have just put the finishing touches to a new Advanced Starter called Test Scores. It is designed to question the misconception that when adding fractions you add both the numerators and the denominators. I hope you get a good reaction from your students who think they have a sound understanding of fraction arithmetic.

The final new addition to the website which appeared last month is called Rough Answers. It is a set of exercises on rounding values in a calculation to find an approximate estimate of the answer. Click on the Description tab to find a link to a Fermi problem about piano tuners. As a Transum Subscriber you have access to a link to a video about Fermi problems and how to solve them. The link is at the bottom of the Fermi Problem page if you are signed in.

I am currently in Bangkok, Thailand where the vast majority of cars have tinted windows so dark that you cannot see anything inside the car. My car does not. Yesterday afternoon I parked my car in the car park next to the Sky Train station and as I got out of the driver’s door I noticed my reflection (these tinted windows act like mirrors) in the window of the car next to mine. I saw that my collar was half up so I straightened it. I also gave my hair a flick then got really close to the window to check I had no vegetation caught between my teeth. Just as I had contorted my face to see clearly my left back molars the engine of the car started and the car pulled away. I felt slightly embarrassed to be honest!

That true story from yesterday is a convoluted way of changing the subject to mirrors and the answer to this month’s puzzle. The time I looked at the clock in the mirror it was 5:15am but appeared to be 6:45am.

That’s all for this edition of the newsletter, I plan to read the new book by Hannah Fry this month called Hello World.

Happy New (School) Year,

John

PS. Maths teachers are very good dancers because they have many algorithms

# Delightfully Divisible

Best wishes for August wherever you may be. I am in the UK and am about to catch a train for Glasgow, a city I’ll be visiting for the first time. The record heat wave in the UK has come to an end and today water is falling from the clouds. I think it is called rain but it has been so long since I saw it that I can’t be sure. Let’s start with a rainy-day puzzle:

Three thousand eight hundred and sixteen is delightfully divisible. The first digit is, of course, divisible by one. The number formed by the first two digits, 38, is divisible by two. The number formed by the first three digits, 381, is divisible by three and the number formed by the first four digits, 3816, is divisible by four.

Can you increase the list of digits to make a nine-digit number which is also delightfully divisible? Your answer should be a pandigital number containing all of the digits one to nine. The answer is at the end of this newsletter.

Talking of Pandigital Numbers, I have just uploaded a brand-new, two-level, self-marking quiz about them which touches on divisibility and, to a lesser extent, place value.

Other activities created in July are based on the ‘arranging the digits 1 to 9’ idea and provide a great environment in which to develop problem-solving strategies. For some the difficulty of the puzzle builds over a number of levels providing a low threshold, high ceiling learning activity. Try them for yourself and please let me have any feedback. They are Multitude, Double Treble and Triside Totals.

Sixteen other Transum activities were updated during last month as part of the Forth Bridge style cycle of keeping all of the content on the website fresh, easy to access and relevant to mathematics learning today. I am always happy to receive comments and suggestions and particularly ideas for new content.

You have probably heard the debate about summer learning loss. Research indicates that by the end of the long summer holidays, pupils perform on average, one month behind where they left off in the spring. It’s not too late to send an email to your pupils with suggestions of Maths activities they can do during the down time. I have put together a list of easy-to-assign activities covering a wide range of topics on the Holiday Activities page. Please let me know if you have any other ideas.

Don’t forget that if ever Transum.org goes offline you can always find the activity you need on one of the mirror sites: Transum.com and Transum.info.

The answer to this month’s puzzle is 381,654,729. Did you enjoy working it out? Would it be a worthwhile challenge for your pupils? Go to the Delightfully Divisible page for an interactive workspace and a link to a list of divisibility tests. Depending on your pupil’s abilities (and the time of day) you may decide to give them a clue as I did to you.

That’s all for now, enjoy the month of August,

John

P.S. Why is a dog with a bad foot like adding 6 and 7?

A. Because he puts down three and carries the one.

# Hot Food and Weather

This is a brief newsletter for the month of July because many of you will have your thoughts focused on the end of term and the holidays. Let’s begin with the puzzle of the month. Can you find five different integers that multiply together to give 12? The answer is at the end of this newsletter.

In the northern hemisphere it’s summer time and here in the UK we are in the middle of an uncharacteristic hot spell. The back garden has become an extension to the house and I am typing this sitting on a plastic chair at a plastic table. The Maths that I think of as I look at the garden is the mathematics of nature. I see Fibonacci numbers in the petals and spirals and remember the great time in the past when pupils have enjoyed the great outdoors doing the Scavenger Hunt and People Maths. This activity suggestion comes with a sun screen and hydration warning!

As I have done a lot of travelling in June there have not been as many new additions to the website as during a typical month but I can tell you about Recipe Ratios, a series of short exercises based on recipes for Thai dishes. They complement the other ratio exercises on the website and come complete with the full method in case you fancied cooking the delicious Siamese food.

Bottles, Boxes and Cans is an activity with two levels. Level one is a drag and drop challenge to match the photograph of a container with its capacity. Level two is a little harder, a more traditional exercise on volume and capacity.

The puzzle of the month is actually the new Starter of The Day for June 12th. It is called Weather Report and puts the puzzle in the context of mean temperatures. The answer is -1, 1, -2, 2, 3.

If this was too easy for you (or your students) there is an extension provided as a new Advanced Starter. What other products of five numbers (less than 100) would have given unique solutions? The answer to that is at the bottom of the Weather Reports page.

I’m off to visit Bletchley Park on Tuesday to see the location of the top-secret codebreaking operation during the Second World War. There is a great amount of mathematics involved in deciphering messages as can be seen in the Code Cracking presentation. I’m sure I’ll have more to say on the topic in next month’s newsletter.

That’s all for now,

John

P.S. Calendars, their days are numbered.

# Square in Rectangle Puzzle

Despite this being the middle of the exam season for some I would still like to take your mind off your daily routine to present you with a puzzle.

Imagine a 10cm by 15cm rectangle. What is the largest square that can be drawn snuggly in one corner that just touches the rectangle’s diagonal. The answer will be at the end of this newsletter.

Now let’s dive into the new activities added to the Transum website this last month.

The Value of Places is, as the title suggests, a place value quiz. It can be sometimes difficult to find more challenging place value activities for Secondary students as they have been learning about place value since early Primary and have probably done the usual types of activity. This online exercise challenges pupils to figure out how many times bigger the value represented by one digit is than the value of another digit in the same number.

Prime Pairs Game is for two players who take it in turns to add a numbered card to either end of a row of cards so that every adjacent pair of cards adds up to a prime number.

Graph Paper has been updated. I think it is the quickest way you can print a sheet of graph paper with numbered axes to suit any graph you may need to draw.

eQuation Generator has also been updated. Its only function is to present you, the teacher, with an endless supply of linear equations that you can project in front of your class. The update improves the delicate balance between providing more of the same while presenting many variations of the chosen type of equation.

Likelihood is a new title given to the probability sorting activity. A pupil’s ordering of the given situations is matched against the average ranking of all the other people who have done the activity.

Loci Land is live but not yet finished at the time of writing this newsletter. I do expect it to be completed in the next week. It currently contains  two real life situations that pupils copy on to squared paper then draw the constructions in order to answer the loci question. More questions will be added soon.

I heard an interesting fact on a podcast I subscribe to last month. Did you know that rather than (in the UK) having 1p, 2p, 5p etc. coins it would be mathematically more efficient to have 1p, 3p, 11p and 37p coins? I have included the excerpt from the podcast in the Transum podcast for this month which you can find at Transum.org/Podcast. If anyone would like to share a proof of that in a form that pupils could understand please let me know and if facts like that are your thing, don’t miss the Maths Trivia page.

On a different subject I am happy to say that you now have the option of viewing Transum webpages via https: as well as http:. This development took place last month when the more secure protocol was implemented. SSL (Secure Sockets Layer) is the standard security technology for establishing an encrypted link between a web server and a browser. This link ensures that all data passed between the web server and browsers remain private and integral. SSL is an industry standard and is used by millions of websites around the world.

Finally the answer to this month’s puzzle is that the area of the biggest square would be 36 square centimetres. You can see the diagram and my method on the ‘Square in Rectangle’ Advanced Starter page.

That’s all for this month,

John

P.S. If a got 50 pence for every time I failed a maths exam I’d have about £6.30 now

# The Royal Game of Ur

Yes, it’s May already and here is the latest Transum Mathematics Newsletter. I hope the year is going well for you and you are finding what you need on the Transum Mathematics website.

The puzzle for this month was inspired by the recent London marathon. Sue Watserface runs the first half at an average speed of 5 miles per hour. What speed would she have to run the second half of the course to attain average speed of 10 miles per hour for the whole race? (answer at end of this newsletter)

The new addition to the website in April that I am most excited about is the Transum version of The Royal Game of UR.

My version is called Remainder Race. Players have to get seven counters around the traditionally shaped board and the number of hops they can make is determined by the remainder left when the number of the square their counter is on is divided by the chosen dice number. In addition to the numeracy skills required there are strategies to be discovered and excitement all round.

It is a game for two players or one player playing against the computer. I am keen to hear how well it works with your pupils. Please let me know.

After writing half of the code for the game on a long-haul plane journey into London, I made it my mission to visit the British Museum and photograph one of the two original game boards. They date from the First Dynasty of Ur, before 2600 BC, thus making the Royal Game of Ur one of the oldest examples of board gaming equipment ever found.

Filling a gap in Transum’s English National Curriculum coverage an online exercise called Furthermore has been created which provides practice for the skill of counting forwards or backwards in steps of powers of ten for any given number up to a million.

Nine Digits is an addition puzzle presented in eight levels of increasing difficulty. You can drag and drop the digits into place. At the time of writing already 78 trophies have been earned for completing the puzzle but no one has yet succeeded beyond level six.

Another new puzzle is called Cubical Net Challenge. Your mission is to paint the faces of the ten cubes represented by their nets. You have two colours, blue and red. No two cubes should look the same no matter which way round the cubes are turned.

Hexblock Hunter is an online game is inspired by TV’s Blockbuster programme (are you old enough to remember that?) and targets lower Secondary/High school mathematical vocabulary. It works best with two players or two teams but the rules can be adapted to create an individual learning activity.

Finally the Privacy Policy has been up dated to specifically address the key features of the new European General Data Protection Regulation (GDPR) legislation which comes into effect later this month.

And now the answer to this month’s puzzle. I was lucky enough to be in London at the time of the marathon. The weather was great for the spectators but a little too hot for the runners.

The length of the London marathon is 26.2 miles but you didn’t need to know that to answer this question. The fact is that the time it would take to run any distance averaging 10mph would have been completely used up by running half that distance at 5mph so only an infinite second half speed would suffice!

Thanks to all those of you who provide feedback, suggestions and ideas for the Transum website. Your messages are very much appreciated.

That’s all for this month.

John

P.S. I don’t trust people with graph paper. They’re always plotting something!

# 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!

# 19 Hidden Gems for Deep Maths Learning

This is the Transum Mathematics Newsletter for February 2018. This month’s puzzle comes from The Penguin Book of Puzzles, a collation of great puzzles from old, out of copyright books written by the prolific puzzle setters from way back.

“Divide 45 in four parts, so that the first part with two added, the second with two subtracted, the third divided by two, the fourth multiplied by two, shall be equal to each other.”

That will give you something to think about. It’s not a familiar puzzle format is it? The answer can be found at the end of this newsletter.

The three most noteworthy new learning objects added to the Transum website this last month are as follows:

Frequency Trees: Being able to construct and read these diagrams is a new topic which appears on both the higher and foundation GCSE(9-1) specifications. This interactive exercise requires pupils to fill in the partially completed frequency trees then try some exam-style questions that introduce calculating probability from the numbers in the trees.

Number Grids: I have started collecting a variety of activities that can be enjoyed using this page of customizable number grids. As a Transum subscriber you get access to buttons that can quickly colour in the grids with the most common number patterns for pupils to describe. Please let me know if you have other ideas for number grid learning activities.

Old Equations: Most students can deepen their mastery of linear equation solving with these old, but just as good as new, intelligently varied questions. These linear equations appeared in a book called A Graduated Series of Exercises in Elementary Algebra by Rev George Farncomb Wright published in 1857.

The book I’m reading at the moment is “What does this look like in the classroom?: Bridging the gap between research and practice” by Carl Hendrick and Robin Macpherson. It is a very readable book in which the current educational experts answer the commonly asked questions about learning backed by the latest research. You can dip in and out of the book focusing on the chapters that interest you. The chapter headings are: Assessment, Marking and Feedback, Behaviour, Reading and Literacy, Special Educational Needs, Motivation, Memory and Recall, Classroom Talk and Questioning, Learning Myths, Technology and Independent Learning.

The thought buzzing around my mind at the moment is the notion that “I think the paradox is that the things that make you a good independent learner don’t necessarily look like independent learning.” Or in other words “…independent learning might be a desired outcome, but paradoxically, it may not be the best way to achieve that outcome.”  Get a copy of the book and read about this and other Maths teacher dilemmas.

Over the last couple of weeks I have been posting “Hidden gems on the Transum website” on Twitter. The idea is to make teachers aware of some of the Transum pages I think are really useful but you wouldn’t necessarily think of searching for. They are also like needles in a haystack as there are over 4000 pages indexed by Google on the Transum website. Here are the gems I’ve already tweeted at the time of writing.

More will be tweeted soon as there are many more hidden gems. You can find out about them if you follow me (@Transum) on Twitter.

The answer to the puzzle of the month is 8, 12, 20 and 5

That’s all for this month,

John

P.S. What do you need to calculate the distance around a circle of sheep?

# 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.

# Christmaths Activities

As December has begun I hope you don’t consider it too early for me to wish you a Merry Christmas. As usual this newsletter will begin with a puzzle of the month. A slightly more difficult puzzle this month that could keep you thinking right through the holiday.

Young Noel Stocking checked his bank account to see how much money he had to buy Christmas presents. When he recorded the balance he wrote down the number of pence for pounds and the number of pounds for pence. A transposition error. “Wahoo” he exclaimed “I’m rich!”

While in this good mood Noel promised 50p to his younger sister Merry. He adjusted his record accordingly which now turns out to be exactly double the amount in the bank.

How much does Noel have in the bank?

Many of the activities on the website have been updated during November and, in preparation for the festive season, the Christmaths collection has been brightly polished. Here are my favourites:

Christmas Ornaments: A puzzle which can be solved online or by using the printable boards so the task can be done in a very practical way with real Christmas ornaments. Problem solving, trial and improvement and logic all wrapped up in one for Christmas.

Christmas Eve Snow: This is a letters-replace-digits puzzle which comes with clues and, if you scroll down to the bottom of the page, a step-by-step guide for solving this kind of puzzle.

Christmas Tables: This is the special Starter for Christmas Day. When I first saw the Betty Edwards drawing I was amazed that the sense of perspective is so powerful. You will need Flash to view the animation.

Cracker Joke: This is a basic numeracy activity where answers to simple calculations are converted to letters to spell out a maths joke. Each time the page loads the calculations change. You can change the joke too.

The Power of Christmas: This Starter works well with Year 11 and Year 12 pupils as it tests their understanding of indices. Finding one solution is fine but the real challenge is to find all four solutions.

The Twelve Days of Christmas: This is a well-worn and time-honoured problem that’s certainly doesn’t deteriorate with age. I particularly like the solution as sung by Natalie Cole.

Christmas Tree Trim: This is just one of the activities on the Transum website that allows pupils to demonstrate their systematic listing skills. There are eight levels of difficulty and a trophy available for each level.

The links to all of these activities (and more) can be found on the Christmaths page. The last week of term is a great time to do some out-of-the-ordinary mathematics with your pupils and there are plenty of ideas on that page. In my experience some of the end-of-term, fun maths has turned out to be the most memorable and enjoyable learning my pupils did all year!

Have your pupils encountered the binary system? There is a visual aid you can use called Binary Lights to demonstrate how binary works. The reason I mention this is that recently I heard a nice idea about counting on your fingers. It’s common knowledge that you can count to ten with the help of fingers and thumbs but if you use binary you can count up to over 1000! Scroll down the Binary Lights page to see a video demonstration.

The answer to this month’s puzzle is £16.33. I found this answer with the aid of a spreadsheet. Please let me know if you have another way of finding this answer.

That’s all for this month

Enjoy the holiday

John

Q. How is an artificial Christmas tree like the square root of minus nine?

A: Neither has real roots!