Building a Puzzle

Play the following two hands with x declared as 3.

Played                                                                                                   Result

                                                    

Record

Maths

Played                                                                                                   Result
                                                   

Record

Maths

Both hands give the same result so

Record

Maths

By not showing what x was declared as we can form a puzzle to find the declared value of x

When we solved

Record

Maths

in The Multiplication Puzzle we took 3x as the played hand and 21 as the result on the board.

Doing the same with 5x + 2 = 3x + 8 we would take 5x + 2 as the played hand
and 3x + 8 as the result on the board.

What does an x piece played on the board look like?

In this case x is declared as 3, so take three black counters and hide them in a box and label the box with an x

                                           
              Counters in box                                                Close box                                    Label box

Now as the wildbox, , contains three black counters make a negative wildbox that contains three white counters, .

Removing the counters gives three pairs of in other words Zero.

So the pair is another form of Zero and can be removed.

TIP In solving this type of puzzle always play cards that will remove wildboxes from the board.

We set up the board by playing a wildbox, , for the wildcard

Having put three black counters in the wildbox this declares the wildcard as a Three.

Set the Puzzle

Played                                                                                               Result

                                                



Swap the hand for the other one that gives the same result which gives the puzzle to play

Puzzle

                                                

Played                                                                                               Result

                

                                              

                  

                                                                           

                               

                                                                         

Record

Maths

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Re-View Mixed Gift Boxes

Let’s take another look at this gift box



Each box contains 3 apples and 4 pears. Applying a bit of maths terminology we could write this as (3 apples + 4 pears ) with the brackets showing that they are boxed together.

There are 5 boxes in the gift box or 5 × (3 apples + 4 pears )

There is a total of 5 × 3 apples and 5 × 4 pears or
5 × (3 apples + 4 pears ) = 5 × 3 apples + 5 × 4 pears = 15apples + 20pears.

Now instead of replacing the apples and pears with known numbers use xs and ys.



In maths this is 5×(3x + 4y) or 5(3x + 4y)

In total there 5×3x s and 5×4y s

or 5(3x + 4y)=5×3x + 5×4y=15x + 20y

In the Game of Maths we use holders rather than boxes.

In cards the following hands give the same result whatever numbers are declared for x and y.

                              5(3x + 4y)

                             5×3x + 5×4y



Wherever there is a gap (or a change of card) above the long holder cut it into card sized holders

Mixed Gift Boxes

A grocer receives two orders for fifteen apples and twenty pears. Using the boxes of apples and pears he has in the shop he puts together one order like this




and one like this



In the first gift box there are five boxes of three apples



And five boxes of four pears



In the second gift box there five boxes containing three apples and four pears

                                    

In both cases there are the same number of apples and pears in the gift box.

Instead of apples and pears we can do the same thing with Threes and Fours.

The Game of Maths

This is the same as arranging a hand of cards in two different ways.

or

and using holders

or which is

In the recording we show that two or more cards held in one holder using brackets.

Record
List           5 x 3 = 15           5 x 4=  20	Maths    5 x 3   +   5 x 4 =15   +   20 = 35

Record
List           5 x (3 + 4) = 5 x 7 = 35                                            35	Maths   5 x (3 + 4) =5 x 7 = 35

We can see that 5 x (3 + 4) = 5 x 3 + 5 x 4

More generally using wildcards

Record
List	Maths

  
         
       

Record
List                             -------------	Maths

We cannot work out a value without being given the declared values for and however we know that   what those values are.

We have found that
is a product of two factors and and we have factorised the number

When dealing with different wildcards think back to The Importance of Counting. Here we saw that all we could do was list ten yachts and twenty crew and could not add the ten and the twenty in any meaningful way.

In the same way because we do not know anything about and and so we must treat them as objects having nothing in common and so we can only put them together as a list