Thursday, November 27, 2014

The Multiplication Puzzle

The multiplication puzzle is one where a record of a game is given but the declared value of a wildcard is not. You have to find the declared value.

For example


3x = 21

Which came from playing this hand where the result is Twenty One.

When you know your times tables it is easy enough to work out that x was declared as a Seven.

For simple puzzles of this sort it is enough to know your times tables to see the value of x straight away.

For more difficult problems we have to find another way. Remember from Why Rules
that simple problems are used to learn the rules of the Game of Maths and not just to write down the answer.

We have used boxes of apples before. Three boxes each containing seven apples.

There are twenty one apples.

We now have the opposite problem.

There are twenty one apples to be shared among, or divided over, three boxes and we want to know how many apples there will be in each box.

Putting one apple into each box in turn shows there are seven apples in each box.

Sharing, or division, undoes the result of multiplication.

Remember that the maths of numbers is about counting and when counting it is very important to be clear that what you are counting have a common type and a common location.

What are we counting in this case are:
Twenty one apples in a pile;
Three boxes in a group;
Seven apples in each box;

Note that in sharing the location of the apples changes and we go from counting the apples in the pile to apples in a box.

In the Game of Maths we are dividing Twenty One equally over three boards and seeing how many on one board.


In this case we are counting:
Twenty one counters in play;
Three boards in the game;
Seven counters on each board;

In the Game of Maths we show a hand for this division as

For which we introduce a new card, the boards cards , showing the number of boards in the game and a new piece the division bar.

The means we are now playing the game with three boards and the hand means share Twenty One equally over three boards.

When playing with many boards we are interested in the results on each board. In practice instead of looking at all the boards we nominate one board as the result board and count what is one this board. We show this board with a wider border.

           Hand                                                                                    Result

In a recording the game we write division in two different ways, either using ÷ or a bar which reflects the way the cards are arranged.

Twenty One divided over three boards is recorded as 21 ÷ 3 or as and we say Twenty One divided by Three or just Twenty One over Three.

Now return to solving the puzzle

          Hand                                                                                    Result
The same result can come from


This result could also be achieved with


Now set up a new game using the       card but with one board for each card used.

As there are now three boards the new game is played with this hand

and these boards

With the result

Each results in so x was declared as Seven

The more you know about multiplication the more you will know about division.

Solving this puzzle using the records

We write

3x = 21

x = 21 ÷ 3 = 7 or x = =7

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