Number Tiles

Task 43 ... Years 4 - 10

Summary

Add two three digit numbers to make a third. The challenge is to do it using only the digits 1 - 9.
  • How many solutions are there?
  • How do you know when you have found them all?
This cameo has a From The Classroom section which tells the story of how a visiting Year 12 student from Uganda used the task and it made him think of another number puzzle. 		 

Materials

  • 9 numbered tiles

Content

  • basic arithmetic skills
  • place value
  • problem solving strategies
  • number patterns
  • connection between addition and subtraction
  • symmetry and combination theory properties
  • symbolic representation
  • concept of proof
Number Tiles

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card. 		   

Finding one solution can take 15 - 20 minutes, and students often ask if they can carry. There is nothing in the problem which says you can't, and there is everything in the problem to indicate it is a normal addition. Carrying introduces an extra 'unstated' digit into the problem, and this is the key to finding a solution, for example:

  249
+318
  567

Five more solutions are:

		 248	 654	 573	 397	 596
		+319	+327	+246	+251	+142
		 567	 981	 819	 648	 738
The iceberg is then driven by the mathematician's questions:
  • How many solutions are there?
  • How do I know when I have found them all?
A clue is that all the answer lines in the solutions to date sum to 18. So, if we can list all the three digit numbers whose digits sum to 18, then we could work backwards to find the pairs of numbers which make each answer line. Applying this strategy of breaking the problem into smaller parts and trying every possible case would then lead to every solution. Classes from Year 5 up have tried this approach by sharing the work between groups.

However, a more rigorous look at this method of solution would notice that it hasn't actually been proved that the answer line digits will always sum to 18. It has only been observed from limited data. There are actually 9x8x7x6x5x4x3x2x1 ways to rearrange the 9 digits into the 9 spaces on the card and without checking them all we can't be sure that one of these doesn't also sum to 18. The software available from the Maths300 lesson below can be used to investigate this question. It also explores all 362,880 rearrangements and confirms that there are 168 solutions - all of which have an answer line that sums to 18.

But is there another way to approach the problem? The answer is yes and here is an outline. Further detail is available in the Maths300 lesson.

All the solutions fit this pattern: 

 abc
+def
 ghi where a, b, ... i are from the set of numbers 1-9
(NB: This doesn't necessarily mean that a = 1, b = 2, etc.)
This is a general representation of the problem because it is the structure of every case. So far there has been no solution without carrying. The following proof shows there isn't one.

First, assume that there is at least one solution which has no carrying. So, given that there is no carrying, adding the columns must give: 

	c + f = i
	b + e = h
	a + d = g
The key now is to realise the unstated fact that:
a + b + c + d + e + f + g + h + i = 45 ... [A]
ie: all nine digits add to 45.

Substituting the first 3 equations into line [A] gives:

2 (g + h + i) = 45
But how can twice a number equal an odd number?

This is a contradiction, so, since there is nothing wrong with the reasoning, the original assumption must be at fault. Therefore, if any solutions exist, they must involve carrying.

Part B of this approach means that a similar exploration must be carried out for these cases:

  • carrying in the ones only
  • carrying in the tens only
  • carrying in both the tens and the ones
Carrying in the ones is shown here as an example. Following through as above, the equations become: 
c + f = i + 10
b + e + 1 = h
a + d = g
which can be rewritten as: 
c + f = i + 10
b + e = h - 1
a + d = g
and then substituting into [A] above gives:
2 (g + h + i) + 9 = 45
hence
2 (g + h + i) = 36
and then
g + h + i = 18
(g + h + i) is the sum of the digits in the answer line so they must add to 18 in this case.
Note: This investigation has been included in Maths At Home. In this form it has fresh context and purpose and, in some cases, additional resources. Maths At Home activity plans encourage independent investigation through guided 'homework', or, for the teacher, can be an outline of a class investigation.

Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works. 		   

To turn this task into a whole class investigation you could make a class set of digit tiles. These would have many other uses. However, you could also ask students to rip up scrap paper so that they made nine pieces between each pair.

Start the problem with a large set of nine digits you prepared earlier. Gather the students in a central place (tables or floor space) and hand out the cards. Ask three students to place their cards to make a 3-digit number. Then ask another three to make a second 3-digit number underneath this:

Now we rule this off here (use a blackboard ruler as the line) and do a normal add up. If we have put the first cards down correctly, the answer will be the tiles left in our hands.
Encourage participation with the group tiles for a while, then promise it can be done and send students to their tables in pairs to see who can be the first team to find a solution with their torn up tiles.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 28, Number Tiles, which also includes companion software.

Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner. 		   

The Number Tiles task is an integral part of:

  • MWA Number & Computation Years 5 & 6

The Number Tiles lesson is an integral part of:

  • MWA Number & Computation Years 5 & 6
  • MWA Number & Computation Years 9 & 10
This task is also included in the Task Centre Kit for Aboriginal Students and the Primary Library Kit. Solutions for tasks in the latter kit can be found here.

From The Classroom

Ugandan Student

Solomon Kyeyune
Year 12 		Semanda Solomon Kyeyune was visiting Australia with his father who is Executive Director of the Rubaga Youth Development Association, Uganda. At nineteen, Solomon had recently completed his Year 12 studies and was waiting university placement. While visiting he explored a couple of tasks, one of which was Number Tiles. A couple of days later he asked:
Would you like to see my trick? I thought of it after using the tile problem.
He explained it this way:

Solomon's Puzzle

Write down any three digit number you like.
Now write another one underneath.
Now I will write one.
Your turn again.
And finally one more from me.

Now I will tell you the sum of these numbers
before you can add them up. It is ...

623
716
283
427
572

 
2621

  • Challenge: Why does it work?
  • What happens if...?
  • Can you invent a similar puzzle?
 		 

Green Line
Follow this link to Task Centre Home page.