The Number Jumbler
The Number Jumbler can always work out your chosen symbol. Can you work out how?
Problem
Take a look at the interactivity below.
Try it a few times.
Can you work out what's going on?
Can you explain how the machine 'guesses' your picture correctly?
Getting Started
Here are the instructions that the Number Jumbler asks you to follow:
Try it a few times with some different starting numbers.
What do you notice about the answer each time?
Student Solutions
We had over 100 submissions from 48 schools in 14 countries, so we're sorry if you or your school are not mentioned here. These examples reflect a cross-section of mathematical thoughts and ideas.
Angel from Kingswells Primary School in Scotland wrote:
I did a few and I was fascinated. How come this machine knows what I'm thinking? Surely it can't be luck... No it isn't. There's a solution to it.
I chose a two-digit number and added the two digits together, then subtracted the total from my original number. Then, instead of pressing the machine I kept the emoji of the answer in my head and did the same thing with another number. I soon found out that the answer to my new number had the same emoji as my first number.
For example:
First number: 18.
18 - (1 + 8) = 9
The emoji for the 9 was a house.
Second number: 43.
43 - (4 + 3) = 36.
The emoji for 36 was also a house.
I did this for a few numbers and the answers were all houses. This shows that the machine made sure that all the answers had the same emoji!
We had many submissions from Westridge School For Girls in the USA (California). Here is just one example:
Now, after a few tries, I found out how the "machine" could always figure out your picture. When the simulation first starts, the pictures seem to be randomly arranged. Most of them are, except one symbol. Every time, one symbol is shown for 9, 18, 27, 36, 45, 54, 63, 72, and 81. This is no coincidence. With the instructions, if you follow them correctly, you should get one of the numbers listed earlier. The program knows this, and it will choose the icon that is shown for 9, 18, 27, 36, 45, 54, 63, 72, and 81.
For example, if the dolphin is the icon for 9, 18, 27, 36, 45, 54, 63, 72, and 81. Since those are the only numbers you can get from following the instructions, the dolphin icon is the only possible picture you could have gotten.
Many solutions from pupils at St. Helens School showed they understood the significance of the 9 times table. Here is an example:
All of the answers we have been getting are in the 9 times tables. All of the numbers between 10 and 19 will end up at 9. This is because if I have 15 that's 15 - 6 so if I have 16 I have added 1 to 15 but it means I have to take an extra 1 away from 16 because it is 1 more away from 9. Also, some of the pictures that are the correct picture are dotted around but this does not mean anything. On one of the rounds we played, I saw that 45 was a frog so I instantly knew that 81 would be a frog too because they are both in the 9x table.
Lots of solutions came from Gospel Oak Primary School. Here is one:
The reason why it can predict what your symbol is, is because your answer is always in the nine times table. If you add two digits together and take that away from your original number it will take you to the nearest multiple of nine. When you add the digits it will give you the distance it is away from the multiples of 9.
Some pupils from St Swithun's Prep School sent in solutions:
First, we found our answer in the chart and memorised our symbol. Then we clicked on the number jumbler machine and it guessed our symbol. Then we repeated the process. We noticed that the symbol's number digit sum was 9.
We then tried all the numbers in the sixties: 60, 61, 62, 63, 64, 65, 66, 67, 68, 69. Each time we got the answer 54. Each time it guessed our symbol but when we did the number 69, we looked at the other numbers with the same symbol as the number 54. We noticed that each number with the same symbol as 54 all had the digit sum of 9.
With this important piece of information, we were able to predict what the symbol would be. In conclusion, the machine always guesses your number because no matter what number on the chart you use, you will always end up with a number that has the digit sum of 9.
David from IES MAXIMO LAGUNA in Spain tried out every single 2-digit number to check that you always get a multiple of 9. Click here to see David's work.
Olivia and her class from Adelaide Botanic High School in Australia found a quicker way to find the result:
To work out which number the Number Jumbler will pick, you need to multiply the first digit by nine.
Here is an example with the number 53:
MY EQUATION:
53 $\rightarrow$ 5 $\times$ 9 = 45
NUMBER JUMBLER WAY:
5 + 3 = 8, 53 $-$ 8 = 45
We had many excellent submissions from Eden Primary School. Here is just one of them:
The solution is that all the multiples of nine until eighty one have the same symbol. Every result will be a multiple of nine. It doesn't guess your number it knows it.
You are probably wondering how it ends up being a multiple of nine. There are two ways to prove this thought. The first one is that there is a pattern with the numbers 99 to 90 - all have the answer of 81 because 99 would be 9 + 9 = 18 and 98 would equal 9 + 8 = 17 so every time the number goes down you subtract less. This also applies for the next set of groups: 89 - 80 = 72 79 - 70 = 63 69 - 60 = 54 and so on the pattern goes.
The second way to prove this thought is if you pick any random two-digit number e.g. 64 and then you add the digits e.g 6 + 4 = 10 you then take that answer you just got (10) and take it away from the number you originally started with so 64 - 10 = 54 which can be divided by 9.
These calculations will prove this method:
49: 4 + 9 = 13 49 - 13 = 36 36 $\div$ 9 = 4
53: 5 + 3 = 8 53 - 8 = 45 45 $\div$ = 5
24: 2 + 4 = 6 24 - 6 = 18 18 $\div$ = 2
Why I chose this solution was because every time I followed the step I noticed it was always a multiple of nine. The fifth time I chose a number I checked the digits and I then found out that all the multiples of nine had the same symbol. I researched more into it and saw that every time it came up with an answer I saw the answer was always the symbol next to nine and its multiple. I noticed that 90 and 99 did not have the same symbol with the nine every time.T his is how I came up with my solution.
Lucas from the Australian International School in Malaysia sent in the following, in which he uses algebra to explain why the answer is always a multiple of 9:
The machine knows the number because no matter what number you choose, the end result will be a multiple of 9. So the machine fills all the multiples of 9 with the same emoji.
It is a multiple of 9 because assuming that the number you chose is N...
N = 10A + B where A is equal to the digit in the tens place and B is equal to the digit in the units place.
A + B = C.
N - C = ?.
Using the substitution method, that means N - C = (10A+B) - (A+B) = 9A.
So since the answer is 9A, it's a multiple of 9.
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem?
The interactivity in this problem performs a trick in which the machine appears to read your mind and pick out the symbol you were thinking of. The mystery of the trick and the amazement when the machine gets it right time after time provide hooks to engage students' curiosity - there is an overwhelming need to explain the mystery and figure out what is going on!
Once students dig a little deeper and apply some mathematical thinking to the situation, the truth emerges - the mathematical steps of the trick force certain properties on the numbers chosen, so all the computer has to do is work within those parameters. Rather than spoiling the magic though, this shows students the power that mathematics has to explain the world, and gives them the tools they need to explore and create similar tricks of their own!
Possible approach
Show the interactivity to the class, and read out the instructions to make sure everyone understands the process:
Ask for suggestions as to how they think the 'trick' might work. Some might think that everyone picked the same number. Others might think it was just lucky. Others might have started to twig that there's some interesting mathematics going on...
Next, try out a few more examples. Students could pick two or three starting numbers and repeat the process. Then collect their answers together on the board. Once there is a conjecture that the answer is always a multiple of 9, return to the Number Jumbler to check whether multiples of 9 share an image.
To complete the understanding of the trick, students could take time to explore why the answer is always a multiple of 9. Here are some possible explanations they might come up with:
"Say you choose a number in the sixties. When you subtract the units, it goes down to 60 and then you take off another 6 for the tens digit which takes you to 54. If you were in the seventies you'd go down to 70 and then 63. I can check them all and it always gives a multiple of 9."
"I chose 34 which is 3 tens and 4 ones. When I added together the two digits and took them away, that's the same as taking away each digit separately. Taking away 4 leaves me with 30 which is 3 tens. Taking away 3 from 3 tens leaves me with 3 nines. It doesn't matter what the digits are, you'll always get a multiple of nine."
"I can use algebra to represent my two digit number. If it has $a$ in the tens column and $b$ in the units column, the number can be written $10a+b$. I then subtract $(a+b)$ from my total. $10a+b-(a+b)=9a$ so the answer is always a multiple of 9.
Key questions
Which numbers in the grid have the same picture?
What do you notice about all the answers you get when you try a variety of starting points?
Why is "add together the digits and subtract the total from your original number" the same as "take away one digit and then take away the other"?
Possible support
A good grasp of place value is really useful for understanding the trick in this problem. Two-digit Targets and Nice or Nasty could be useful for consolidating place value before working on the problem.