| T | H | R | E | E | |
| T | H | R | E | E | |
| + | F | O | U | R | |
| E | L | E | V | E | N |
Can you find digits to replace the letters to make this sum correct?
Note to teacher: If you are presenting this puzzle to your class using a projector you can slowly scroll down the step by step guide below giving clues that might enable your students to finish off the puzzle for themselves.
There is no standard way to solve a problem like this. Each one is quite different in structure and will have its own clues; but having said that here is an example of how to solve this particular challenge which may give you ideas for solving others.
We should assume that each letter stands for a different digit.
1. The first thing to notice is that the bottom line has six digits. When the two Ts are added together the answer caused a digit to be carried. This carried digit is most likely a one though there is a small possibility it could be a 2. Lets go with one and backtrack to here if it does not work out.
| T | H | R | 1 | 1 | |
| T | H | R | 1 | 1 | |
| + | F | O | U | R | |
| 1 | L | 1 | V | 1 | N |
2. Looking at the tens column, to produce a digit one in the bottom line, the U must be a nine unless there is a digit carried from the units column. Lets go with nine and backtrack to here if it does not work out.
| T | H | R | 1 | 1 | |
| T | H | R | 1 | 1 | |
| + | F | O | 9 | R | |
| 1 | L | 1 | V | 1 | N |
It may be useful to keep a record of the letters involved in the puzzle and the possible digits they could stand for:
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
| T | ✗ | ✗ | ||||||||
| H | ✗ | ✗ | ||||||||
| R | ✗ | ✗ | ||||||||
| E | ✗ | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ |
| F | ✗ | ✗ | ||||||||
| O | ✗ | ✗ | ||||||||
| U | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✓ |
| L | ✗ | ✗ | ||||||||
| V | ✗ | ✗ | ||||||||
| N | ✗ | ✗ |
3. Now it is a little more difficult to spot any certainties though we can see that if R is even then N is also even (and the inverse).
Let's make a guess and see how far it takes us. We'll start with the units column as there won't be any carried numbers to worry about.
Our guess for R will be the smallest digit it could possibly be, zero. If that is not correct and we get stuck further along we can always backtrack to this point and change our guess to the next smallest digit and so on.
If R is zero then N must be 2.
At this point be careful you don't get the letter O and the digit zero confused. We will use the slashed zero, Ø, to avoid confusion.
| T | H | Ø | 1 | 1 | |
| T | H | Ø | 1 | 1 | |
| + | F | O | 9 | Ø | |
| 1 | L | 1 | V | 1 | 2 |
The table is updated with this information.
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
| T | ✗ | ✗ | ✗ | ✗ | ||||||
| H | ✗ | ✗ | ✗ | ✗ | ||||||
| R | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ |
| E | ✗ | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ |
| F | ✗ | ✗ | ✗ | ✗ | ||||||
| O | ✗ | ✗ | ✗ | ✗ | ||||||
| U | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✓ |
| L | ✗ | ✗ | ✗ | ✗ | ||||||
| V | ✗ | ✗ | ✗ | ✗ | ||||||
| N | ✗ | ✗ | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ |
By working through the columns from right to left we can deal with the carried numbers as they appear.
4. Let's make another guess. This time for the letter O. We can see from the table that the smallest digit it could possibly be is a three. If that is not correct and we get stuck further along we can always backtrack to this point and change our guess to the next smallest digit and so on.
If O is 3 then V must be four as one is carried over from the 10s column.
| T | H | Ø | 1 | 1 | |
| T | H | Ø | 1 | 1 | |
| + | F | 3 | 9 | Ø | |
| 1 | L | 1 | 4 | 1 | 2 |
The table is updated with this information.
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
| T | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ||||
| H | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ||||
| R | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ |
| E | ✗ | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ |
| F | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ||||
| O | ✗ | ✗ | ✗ | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ |
| U | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✓ |
| L | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ||||
| V | ✗ | ✗ | ✗ | ✗ | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ |
| N | ✗ | ✗ | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ |
5. Let's make another guess. This time for the letter H. We can see from the table that the smallest digit it could possibly be is a five.
If H is five, F would have to be one to produce the given column total but we have already assigned the digit one so H cannot be five. Let's backtrack.
We can see from the table that the next smallest digit it could possibly be is a six.
If H is six, F would have to be nine to produce the given column total but we have already assigned the digit nine so H cannot be six. Let's backtrack.
We can see from the table that the next smallest digit it could possibly be is a seven.
If H is seven, F would have to also be seven to produce the given column total but we cannot assign the same digit to two different letters. Let's backtrack.
We can see from the table that the only other possible choice for the letter H is eight.
If H is eight, F would have to be five:
| T | 8 | Ø | 1 | 1 | |
| T | 8 | Ø | 1 | 1 | |
| + | 5 | 3 | 9 | Ø | |
| 1 | L | 1 | 4 | 1 | 2 |
The table is updated with this information.
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
| T | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ||
| H | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✓ | ✗ |
| R | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ |
| E | ✗ | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ |
| F | ✗ | ✗ | ✗ | ✗ | ✗ | ✓ | ✗ | ✗ | ✗ | ✗ |
| O | ✗ | ✗ | ✗ | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ |
| U | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✓ |
| L | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ||
| V | ✗ | ✗ | ✗ | ✗ | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ |
| N | ✗ | ✗ | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ |
6. Let's now guess the letter T. We can see from the table that the smallest digit it could possibly be is a six.
Two sixes with the two carried from the thousands column makes 14 but L cannot be four as that digit has already been assigned. Let's backtrack.
The only remaining possibility is that T is seven making L equal to six.
| 7 | 8 | Ø | 1 | 1 | |
| 7 | 8 | Ø | 1 | 1 | |
| + | 5 | 3 | 9 | Ø | |
| 1 | 6 | 1 | 4 | 1 | 2 |
Here are some other solutions found by making different guesses in the process above. Can you find any others?
| 8 | 4 | Ø | 1 | 1 | |
| 8 | 4 | Ø | 1 | 1 | |
| + | 3 | 5 | 9 | Ø | |
| 1 | 7 | 1 | 6 | 1 | 2 |
| 7 | 4 | 6 | 1 | 1 | |
| 7 | 4 | 6 | 1 | 1 | |
| + | 2 | Ø | 9 | 6 | |
| 1 | 5 | 1 | 3 | 1 | 8 |
| 4 | 6 | 5 | 1 | 1 | |
| 4 | 6 | 5 | 1 | 1 | |
| + | 8 | 2 | 9 | 5 | |
| 1 | Ø | 1 | 3 | 1 | 7 |
Here are some other word sums to try:
Finally here is the URL which will take you to a code cracking activity.
Christmas Present Ideas
It is often very difficult choosing Christmas presents for family and friends but so here are some seasonal, mathematics-related gifts chosen and recommended by Transum Mathematics.
Go Genius MathsTransform maths practice into an exciting adventure with Go Genius Maths! This award-winning board game is designed to make learning maths enjoyable and engaging for children aged 7 and above. Perfect for family game nights or classroom activities, Go Genius Maths helps players develop essential arithmetic skills while having heaps of fun. Whether you’re looking for a gift to inspire young minds or a tool to supplement maths education, Go Genius Maths is the ultimate choice. Spark curiosity, nurture mathematical talent, and turn learning into an unforgettable journey! |
|
||||
|
How Not To Be WrongThe maths we learn in school can seem like an abstract set of rules, laid down by the ancients and not to be questioned. In fact, Jordan Ellenberg shows us, maths touches on everything we do, and a little mathematical knowledge reveals the hidden structures that lie beneath the world's messy and chaotic surface. In How Not to be Wrong, Ellenberg explores the mathematician's method of analyzing life, from the everyday to the cosmic, showing us which numbers to defend, which ones to ignore, and when to change the equation entirely. Along the way, he explains calculus in a single page, describes Gödel's theorem using only one-syllable words, and reveals how early you actually need to get to the airport. What more could the inquisitive adult want for Christmas? This book makes a cosy, interesting read in front of the fire on those cold winter evenings. more... #ad |
||||
Graphic Display CalculatorThis handheld device and companion software are designed to generate opportunities for classroom exploration and to promote greater understanding of core concepts in the mathematics and science classroom. TI-Nspire technology has been developed through sound classroom research which shows that "linked multiple representation are crucial in development of conceptual understanding and it is feasible only through use of a technology such as TI-Nspire, which provides simultaneous, dynamically linked representations of graphs, equations, data, and verbal explanations, such that a change in one representation is immediately reflected in the others. For the young people in your life this is a great investment. Bought as a Christmas present but useful for many years to come as the young person turns into an A-level candidate then works their way through university. more... #ad |
|
||||
|
Apple iPad ProThe analytics show that more and more people are accessing Transum Mathematics via an iPad as it is so portable and responsive. The iPad has so many other uses in addition to solving Transum's puzzles and challenges and it would make an excellent gift for anyone. The redesigned Retina display is as stunning to look at as it is to touch. It all comes with iOS, the world's most advanced mobile operating system. iPad Pro. Everything you want modern computing to be. more... #ad Before giving an iPad as a Christmas gift you could add a link to iPad Maths to the home screen. |
||||
Craig Barton's Tips for TeachersTeaching is complex. But there are simple ideas we can enact to help our teaching be more effective. This book contains over 400 such ideas." more... #ad "The ideas come from two sources. First, from the wonderful guests on his Tips for Teachers podcast - education heavyweights such as Dylan Wiliam, Daisy Christodoulou and Tom Sherrington, as well as talented teachers who are not household names but have so much wisdom to share. Then there's what he has learned from working with amazing teachers and students in hundreds of schools around the world. |
|
||||
|
The Story Of Maths [DVD]The films in this ambitious series offer clear, accessible explanations of important mathematical ideas but are also packed with engaging anecdotes, fascinating biographical details, and pivotal episodes in the lives of the great mathematicians. Engaging, enlightening and entertaining, the series gives viewers new and often surprising insights into the central importance of mathematics, establishing this discipline to be one of humanity s greatest cultural achievements. This DVD contains all four programmes from the BBC series. Marcus du Sautoy's wonderful programmes make a perfect Christmas gift more... #ad |
||||
Christmas MathsThis book provides a wealth of fun activities with a Christmas theme. Each photocopiable worksheet is matched to the Numeracy Strategy and compatible with the Scottish 5-14 Guidelines. This series is designed for busy teachers in the late Autumn term who are desperate for materials that are relevant and interesting and that can be completed with minimun supervision. All the activities are suitable for use by class teachers, supply teachers, SEN teachers and classroom assistants and cover topics such as 'How many partridges did the true love give all together?' and 'Filling a sleigh with presents by rolling a dice!'. Children will have lots of fun working through the Christmas Maths themes but also gain valuable skills along the way. A great source of ideas and another reasonably priced stocking filler. more... #ad |
|
||||
|
A Compendium Of Mathematical MethodsHow many different methods do you know to solve simultaneous equations? To multiply decimals? To find the nth term of a sequence? A Compendium of Mathematical Methods brings together over one hundred different approaches from classrooms all over the world, giving curious mathematicians the opportunity to explore fascinating methods that they've never before encountered. If you teach mathematics to any age group in any country, you are guaranteed to learn lots of new things from this delightful book. It will deepen your subject knowledge and enhance your teaching, whatever your existing level of expertise. It will inspire you to explore new approaches with your pupils and provide valuable guidance on explanations and misconceptions. more... #ad |
||||
Math with Bad DrawingsI had been tutoring the wonderful Betsy for five years. When the day came for our last ever session together before the end of her Year 13, I received this beautiful book as a gift of appreciation. This a very readable book by Ben Orlin. I'm really enjoying the humour in the writing and the drawings are great. Ben Orlin answers maths' three big questions: Why do I need to learn this? When am I ever going to use it? Why is it so hard? The answers come in various forms-cartoons, drawings, jokes, and the stories and insights of an empathetic teacher who believes that mathematics should belong to everyone. more... #ad |
|
||||
Click the images above to see all the details of these gift ideas and to buy them online.
Your access to the majority of the Transum resources continues to be free but you can help support the continued growth of the website by doing your Amazon shopping using the links on this page. Below is an Amazon link. As an Amazon Associate I earn a small amount from qualifying purchases which helps pay for the upkeep of this website.
Educational Technology on Amazon
Do you have any comments? It is always useful to receive feedback and helps make this free resource even more useful for those learning Mathematics anywhere in the world. Click here to enter your comments.
Transum,
Monday, January 20, 2020
"The largest calculation of this type I have come across is the Poco Poco Puzzle. How many fews make a lot in Spanish? It first appeared in the Puzzle Corner, MIT News, September 2014:
P O C O
P O C O
P O C O
P O C O
P O C O
P O C O
P O C O
P O C O
P O C O
P O C O
P O C O
P O C O
P O C O
P O C O
+ P O C O
= M U C H O
Luckily the solution was beautifully explained in the Guardian in January 2020."