Sicherman DiceTask 241 ... Years 4  10SummaryDiscovered 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 nonzero 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
Content

IcebergA 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.
When you count the number of ways each outcome can be made the table becomes:
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, nonzero 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. 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!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.
Again, when you count the number of ways each outcome can be made the frequency table remains the same.
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.
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. ExtensionsRoy 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:
I remain convinced that there is no solution for a 15 dice but do let me know if one is found. Occasionally I still exercise my obsession and attempt to find a solution for a 18 dice but it is proving most difficult to track.

Whole Class InvestigationTasks 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.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 (nonzero) 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 handson learning kits available from Years 310 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. 