Task 190 ... Years 4 - 10
SummaryAs the history on the card suggests, this puzzle has been well know to mathematicians. In part, the reason is that it is difficult and Mathematics just love a challenge. (a quote from Andrew Wiles in Fermat's Last Theorem, a BBC Horizon documentary). Hints are given on the card and it is important that they are covered, as instructed. The fewer hints students use in the solution, the more satisfaction they will have when solving it. But don't expect that solution to happen in one session.
Related tasks using the same reasoning strategies are:
IcebergA task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.
This is probably the most difficult of the 'magic' tasks in our task collection. To begin the problem, the way of thinking is the same as for the other tasks above:
However, the difference between this and the other magic problems is that there are not the same number of addends in each partial sum. In some columns there are three; in some four; in some five.
The three directions involved in the puzzle are also spatially unusual.
As far as is known there is only one solution to this problem - although there are several symmetry transformations of this solution. So the value of this task is the application of problem solving strategies, rather than the extensions which might be developed. For example, breaking the problem into smaller parts shows that there is an outer ring of six sets of three-digit numbers which sum to 38. Further, each set overlaps with the one clockwise and anti-clockwise from it at one number. This insight leads to applying the 'try every possibility' strategy to list all the sets of three numbers (from 1 - 19) which sum to 38.
So Size 1 and Size 3 cannot be magic, but Size 5 is magic.
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.
Possibly the best way to turn this task into a whole class lesson is within a unit which includes Magic Squares, Fraction Magic Square and Magic Cube. These all highlight breaking a problem into parts and trying every possible case. Magic Hexagon can be the challenge near the end of the unit to see how well skills are transferred to a new situation. As suggested above, it could be a paper and pencil exercise, but it is far less frustrating to use counters. You need a set numbered from 1 - 19 for each pair, and it doesn't take long to produce them. Each pair also needs a hexagon grid.
Present the problem as a possible outcome of a mathematician successfully working with Magic Squares and asking What happens if we have a different shape of grid?. Could we make it magic? Ask the students to tell you how many digits would be needed if the problem was going to work and then hand out the sets of counters and the grid.
What did we learn from Magic Squares that might start us off in this investigation?Continue from here being careful to minimise the clues you give and to maximise the opportunities for sharing and peer teaching across the class.
At this stage, Magic Hexagon does not have a matching lesson on Maths300.
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 Magic Hexagon task is an integral part of: