The Riddle of the Sphinx

This post is about an activity that helps students practice recognizing which numbers are factors of others. It is intended for grade five and above. The activity has several different forms and we will comment on which are harder as each variant is described. In order to run this activity, the parent or teacher will play the part of the Sphinx. If you have an Egyptian head dress or other prop, that helps set the mood.

The game starts with the Sphinx laying out cards like the ones below and saying

“This is my riddle. Choose a card and I will tell you whether you may take it or not. If you take a card, I will take others. If there are no cards left that you may take, then the remaining cards come to me. Your task is to take cards to gain the largest sum. To discern the rules by which I take cards and allow cards is your true problem. Three times may you attempt this task, then done!”

If you don’t have cards, you can write the numbers on a piece of paper, circling the students choices and crossing out those that the Sphinx takes. Here are the actual rules:

1. For the student to be allowed to take a card, there must be another card left whose value divides it evenly, e.g., to take 10 one of 1, 2, or 5 must be left.
2. When the student takes a card, the Sphinx takes all the cards that evenly divide it.
3. If there are no cards left the student can take, the Sphinx gets any remaining cards.
4. “Three times …” means the student can start over three times.

Alternatively, the Sphinx may let the student know the rules from the beginning. Oddly, experience suggests that younger students will be more comfortable with just wading in without the rules. Occupy Math thinks this is less fun, but it may also be less intimidating. This activity can be part of discovery learning to discover what a divisor is. By increasing the size of the largest card you can create a whole family of puzzles.

This puzzle is pretty easy if the largest card is at most eleven. The smallest version of the puzzle that has any strategy is the one with largest card six. It gets harder at twelve and keeps getting harder. Occupy Math would say 7th or 8th grade for twelve, 9th or 10th for twenty, and 11th and 12th for twenty-four. The version with largest number ten is good for learning the game. As students master one version, increase the size by one or two. If they work on it, younger students may well be able to master harder versions.

The general answer, in the form of the best strategy, is not known. Occupy Math wrote a computer program that plays all possible games to find the best possible score for the versions of the puzzle from six to twenty-four. They are listed below with a sequence of cards the student takes to get that score. There are often many ways to get the best score. There is one universal rule known: always take the largest prime number first. If a student deduces this rule, praise them.

The following giant list is a best-possible solution to the activity for six through twenty-four cards. The answers give the size of the largest card in letters, then the best possible score. After the colon is one possible sequence of cards the student can choose — in their correct order — to get that best score.

Easy versions of the activity.
Six, 15 : 5 4 6
Seven, 17 : 7 4 6
Eight, 21 : 7 6 8
Nine, 30 : 7 9 6 8
Ten, 40 : 7 9 6 10 8
Eleven, 44 : 11 9 6 8 10

Intermediate versions of the activity.
Twelve, 50 : 11 10 8 9 12
Thirteen, 52 : 13 10 8 9 12
Fourteen, 66 : 13 10 14 8 9 12
Fifteen, 81 : 13 9 15 10 8 12 14
Sixteen, 89 : 13 9 15 10 12 14 16
Seventeen, 93 : 17 9 15 10 12 14 16
Eighteen, 111 : 17 9 15 10 14 18 12 16
Nineteen, 113 : 19 9 15 10 14 18 12 16

Hard versions of the activity.
Twenty, 124 : 19 10 14 15 20 12 16 18
Twenty one, 144 : 19 9 15 21 14 18 12 16 20
Twenty two, 166 : 19 9 15 21 14 18 12 16 20 22
Twenty three, 170 : 23 9 15 21 14 18 12 16 20 22
Twenty four, 182 : 23 9 15 21 14 16 18 20 22 24

What else can be done with this activity?

• The game can be played as a solitaire where the student is also the Sphinx.
• There is another theme for this activity called the Tax Man game. Follow the link to our sister blog, Dan and Andrew’s Game Place.
• Trying to get the lowest possible score is also difficult.
• An alternate scoring system is to find a way to take the most (or least) cards.

The activity in the post is primarily intended to make learning about and practicing factoring and recognizing factors more interesting. The Egyptian motif is not necessary, but Occupy Math thinks a little more showmanship might make math more palatable. It is not hard to come up with other skins for this activity or different activities based on it. It you find something good (or want to know the low scores), please comment or tweet!

I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics