Magic Dice

IWB DiceOne of the ways in which this site will provide rich resources for the classroom is to plunder the appetite for puzzles that society before television had.  These puzzles are found in many books and on many sites as they are now out of copyright but they are not often packaged in a way ready for use by teachers.  While teachers can source and present these for themselves it is something that there is rarely time for. 

Here is the first one courtesy of the fantastic English puzzle master Henry Ernest Dudeney who died in 1930 but left a wealth of puzzles that deserve to see the light of day again.

This is a particularly nice puzzle as it starts with a hint of magic and children of all levels can unearth something of its mysteries.  At its simplest level children of all abilities will be able to learn how to carry out the trick.  Those that are more able can look at the mathematical manipulation  and see the some or all of the mechanism by which it works.  Those that master this could even be introduced to algebra using this trick.

Resources required:

Three actual dice or the Space Dice resource on teacherled.com.

A good way to first introduce this lesson is to stand with your back to the IWB and have a student press the button to roll the three dice on the IWB resource.  Making it clear that you are not able to see the result – ask the class to do the following. 

Step 1: Multiply the first dice by 2

Step 2: add 5

Step 3: Multiply this result by 5

Step 4: Add the second value of the second dice

Step 5: Multiply the result by 10

Step 6: Add the points of the 3rd dice

Using the picture above as an example the class would now tell you the number 595.  In an instant you tell them the results of the three dice.

Their challenge is to 1) be able to duplicate the trick 2) explain how it works.

Here is a selection of printable cards that you can give to the students.  You can also give them 3 dice each or they can make up 3 results to test their ideas.

To duplicate the trick they will need to discover that taking 250 off the result gives you the result.  From the example above: 595 – 250  = 345 which is the results on the dice.

In algebra let the 1st dice be a the 2nd b and the 3rd c

Step 1:  Multiply the first dice by 2 –> 2a

Step 2: Add 5  –> 2a+5

Step 3 Multiply by 5 –> 10a + 25

Step 4:Add the 2nd dice  –> 10a + 25 + b

Step 5: Multiply by 10 –> 100a + 250 + 10b

Step 6: add the 3rd dice –> 100a + 250 + 10b + c

From this you can see that the first dice is multiplied by 100 to give it the place value in the 100s column, b is multiplied by 10 to put it in the 10s column and c not multiplied leaving it in the units column.  And then there is 250 generate in steps 2, 3 and 5- obfuscating the result.

In other words the result you get from following the instructions is 100 of the 1st dice + 10 of the 2nd dice + 1 of the 3rd dice plus 250.

If you have enough physical dice (110!) you could actually lay them out showing the multiplication of them.  Ensure that they see the 250 is in addition to the dice – record it as the number on paper in aside from the dice.

As an extension more able students, if they understand this mechanism, they could design their own set of instructions to create a similar trick.

When you judge the students could do with a hint these suggestions will give them useful clues.

Discovering the trick is to subtract 250

Working out from the instruction where the 250 comes from

Why the 1st dice is shown by the 100s column, the 2nd by the 10s column and the 3rd by the units.