Magic Hexagon

Task 190 ... Years 4 - 10

Summary

As 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:

Materials

Playing board and 19 counters numbered 1 to 19

Content

arithmetic, addition / subtraction
arithmetic, multiplication / division
average
history of mathematics
mental arithmetic
reasoning

Iceberg

A 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:

What it the grand total of the available digits?
Looking in just one direction how many ways does this total have to be shared?

So, in this case, the digits 1 - 19 sum to 190 and they have to be shared between 5 vertical columns of the hexagon. Therefore each column has to sum to 38.

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.

38 = 1 + 37 = 1 + 19 + 18
38 = 2 + 36 = 2 + 19 + 17 (1 + 18 + 18 is not allowed - digits are not repeated)
38 = 3 + 35 = 3 + 19 + 16 = 3 + 18 + 17
38 = 4 + 34 = 4 + 19 + 15 = 4 + 18 + 16
38 = ...

Eventually, all sets of all triples which sum to 38 will be found. Then by:

selecting sets of six triples, and
shuffling numbers within triples so that the 'tail' of one triple is the same as the 'head' of the next

the outer ring can be found. This is far easier, and in some ways more rewarding, to do with the numbered discs than it is by pencil and paper.

Now the problem looks a little easier. The numbers remaining are 1, 2, 4, 5, 6, 7, 8 and it might be worth trying the middle number in this set in the middle of the shape. Eventually, with as little hint as possible, students will discover the solution.

One further investigation relates to seeing this diagram as a 'nest' of potential magic hexagons.

The central single hexagon would be Size 1. Nothing very magic about that.
The Size 3 (named by its longest chain of unit hexagons) would be...

Would the numbers 1 - 7 make a magic hexagon in the Size 3? A worthwhile question, but It only takes a moment to realise that the sum of these digits is 28 which can't be divided by 3 to produce integer results.

So Size 1 and Size 3 cannot be magic, but Size 5 is magic.

Is there another Size of nested hexagons which might form a magic hexagon?
If there is, can the magic hexagon be made?
If we can't find another possibility, can it be proven that there are none?

Investigating these questions involves working out:

size number
number of unit hexagons - which determines the upper limit of the consecutive numbers starting from 1
finding the total of the sequence 1 to ...
deciding the number of columns the total is shared between
checking whether dividing by this number gives a whole number answer

This exploration is continued in Maths With Attitude, Number & Computation, Years 9 & 10 and triangle numbers play a part.

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.
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:

MWA Number & Computation Years 9 & 10

Follow this link to Task Centre Home page.