Sicherman Dice

Task 241 ... Years 4 - 10

Summary

Discovered by George Sicherman and recorded in a Scientific American article in 1978 these specially numbered cube dice give the same table of outcomes as a pair of standard 1 - 6 dice. Easy to say, but it took a (recreational) mathematician to ask What happens if..? for them to be discovered. The task allows students to make their own (guided) discovery of the defining property of Sicherman Dice using the only whole number non-zero dice set that works. The challenge then has more solutions when the students are allowed to use zero on the dice faces? ...or what happens if negative numbers are involved?

This cameo includes an Investigation Guide.

 

Materials

  • Two standard cube dice
  • Two special cube dice marked 1,2,2,3,3,4 and 1,3,4,5,6,8

Content

  • arithmetic, addition / subtraction
  • concept of proof
  • probability calculations
  • probability experiences
  • probability, sample space / sample size
  • statistics, analysing data
  • statistics, collecting & organising data
  • statistics, frequency
Sicherman Dice

Iceberg

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

 

Sample Space showing all possible outcomes of rolling and adding two dice.

+ 1 2 3 4 5 6
1 2 3 4 5 6 7
2 3 4 5 6 7 8
3 4 5 6 7 8 9
4 5 6 7 8 9 10
5 6 7 8 9 10 11
6 7 8 9 10 11 12

When you count the number of ways each outcome can be made the table becomes:

Poss. Totals 2 3 4 5 6 7 8 9 10 11 12
No. of Ways 1 2 3 4 5 6 5 4 3 2 1

No. of ways is also called Frequency.

   

Possibly the best way to explain this task is to tell the story of how it came to us. When Matthew Reames was teaching at St. Edmund's Junior School, England, he journeyed through many investigative challenges with his class. Examples are recorded in Task 45, Eric The Sheep, and Task 211, Soft Drink Crates. Then one day he wrote:

I have been looking at Sicherman Dice recently (these dice are evidently the only other set of dice using positive, non-zero integers to give the same possible outcomes as a 'regular' pair of dice). Rather than being the 'normal' 1 to 6, these are two different dice, one numbered 1, 2, 2, 3, 3, 4 and the other numbered 1, 3, 4, 5, 6, 8.

I used these with Year 6 and 7 classes (aged about 10 and 11 this time of year). In future lessons I would probably include this in the regular probability things, but this was one of those end-of-term lessons. We talked about the possible outcomes of flipping one coin, of rolling one number cube, then flipping two coins, and finally rolling two 'normal' dice. We made an outcome grid for addition (similar to the one on the Dice Differences task cameo but with addition rather than subtraction).

Then, I asked the class to find another set of dice that would give the same possible outcomes (not necessarily in the same order) on the chart - we talked about why. When allowed to use zero or negative numbers, there are a huge (possibly infinite) number of possibilities. There was a lot of good discussion about this!

Then, I said that they could not use zero or negative numbers (or fractions or decimals, etc - only integers). Though this required a bit more guidance from me (and I eventually had to tell them the final combination), there was good discussion about what they knew some of the numbers had to be (for example, each cube had to have a 1 otherwise they could not make a 2).

Taking Matthew's point here, we decided in preparing a task that was intended to be used by a pair of students relatively independently that asking them to rediscover Sicherman's dice was a big call for most. Hence the scaffold presentation on the card that gives the dice values and requires the students to discover that these two special dice result in the same frequency table.

+ 1 3 4 5 6 8
1 2 4 5 6 7 9
2 3 5 6 7 8 10
2 3 5 6 7 8 10
3 4 6 7 8 9 11
3 4 6 7 8 9 11
4 5 7 8 9 10 12

Again, when you count the number of ways each outcome can be made the frequency table remains the same.

Possible Totals 2 3 4 5 6 7 8 9 10 11 12
Frequency 1 2 3 4 5 6 5 4 3 2 1
All of this was leading up to some further activities on the TI Nspire handhelds we were borrowing - talking about probability distribution and why, for a small number of rolls, the frequency graphs they made were so different but then got more and more identical as they increased the number of rolls.

Anyway, I wasn't sure if you already had a task similar to this or not, but I thought it was worth passing along just in case. My classes certainly had some fun trying to figure it out and there was a lot of excellent maths discussion along the way.

Matthew has also supplied this spreadsheet which has two examples of whole number dice with zero as one face. (Note: The spreadsheet has been zipped.)

Once students understand how these tables are made, you might want them to search for more pairs of dice with the same sample space (all possible outcomes) as the 'normal dice'. In doing so you might want them to make the tables by hand because of the arithmetic practice, or you might, as Matthew did, want to introduce them to the idea of using a spreadsheet as a tool.

Can you see how these two dice were created
from 'normal dice'?

Matthew's spreadsheet is an electronic Investigation Guide. You enter the dice numbers in the top row and left column and the sheet contains formulae which, for any cell, automatically sum the left column dice number for that cell with the top row dice number for that cell. It makes it very quick to try out hypotheses about the dice numbers.

Also, given that we are aiming for the sums 2 and 12 to occur only once in any table, if the appropriate numbers are put first and last in the column and row, then only four other numbers need to be found for each dice.

Matthew's sheet contains examples his class developed using whole numbers, decimals, fractions and negative numbers. Wow! Now there's millions of solutions.

Extensions

Roy Grice learnt about the Sicherman Dice challenge through the February 2013 Mathematics Centre eNews when we introduced the companion Maths300 lesson to this task. All Roy knew from the brief news item was that the challenge was to 'design two new cube dice which will produce the same frequency table' as the one for 1- 6 cube dice.

At the April annual Easter conference of ATM he passed on his thanks for the problem - he had thoroughly enjoyed the search - and offered these extra challenges:

  • What happens if we use four sided dice?
  • Are there any other dice with the Sicherman property?
When these challenges were offered later in April at a meeting of SMaL-Syd, Sweden, a couple of days later, one teacher offered one solution to the first of them. Unfortunately, neither the teacher's name, nor the solution is recorded in our correspondence.
  • Can you rediscover that solution?
  • How many solutions are there for the four sided dice?
Roy is sure that there are no solutions to the case of a five sided dice. In Roy's own words from an email in May 2013:
I remain convinced that there is no solution for a 1-5 dice but do let me know if one is found. Occasionally I still exercise my obsession and attempt to find a solution for a 1-8 dice but it is proving most difficult to track.
  • Anyone else willing to continue join these investigations?

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. 		   

Matthew has given us the outline of a whole class investigation above. The use of the spreadsheet in conjunction with this number and probability investigation is great example of connected learning in mathematics. He comments:

I let the kids work on finding some possibilities (they could use fractions, decimals, zero and negatives) and they came up with quite a few possibilities and explained how they come up with them and why there are so many possibilities. Then, I limited the them to only positive integers. This was quite a bit more difficult.

They um'ed and ah'ed for a while until they realised that each cube had to have the number 1 (otherwise they couldn't have made a 2 in the grid). They then realised that the highest possible number on one cube was 11 (at this point it was not certain if it was 11 but they knew it couldn't be any higher than 11 otherwise they would have gotten numbers larger than 12 in the grid). At some point, I gave them a couple more numbers from the dice and let them work further. Eventually they got them all.

During all this, I had a spreadsheet projected on the whiteboard so we could immediately test their ideas and suggestions for the numbers.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 185, Sicherman Dice which introduces the lesson by reconnecting students with the table of outcomes for 1 - 6 cube dice and goes on to focus on the strategies that could be used to find the Sicherman Dice numbers. In particular, since there is only one possible (non-zero) way to make the sum 2, it uses the starting point of trying all possible ways of making the sum 12. Then for each of these, examining the remaining possibilities for the four other numbers on each dice.

As Matthew's commentary suggests, it is a joyful lesson that has students captivated and fascinated by, and totally absorbed in, working like a mathematician.

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. 		   

Sicherman Dice is not in any MWA kit. However it can be used to enrich the Chance & Measurement or Number & Computation kit at Years 5/6 or 9/10.

Green Line
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