## Sunday, November 22, 2009

### Prime number monopoly

By popular request from an awesome teacher friend, here are directions for Prime Number Monopoly, a fun game that I taught to teachers attending the New York City Math Teacher Circle retreat last summer. This is also a great game for older students to teach their younger siblings. Three students makes a good-size group for this game, but you can play with two or four per group as well.

There's minimal equipment involved. Each group needs a set of deeds (see below), a pair of dice, and paper and pencil for keeping track of scores. That's it. No playing board is needed. No play money is needed either, because students keep track of their balances as a running account.

Start by creating a set of deeds for the prime numbers for each group of students. You can probably get your students to figure out my pricing scheme from the picture of the first three deeds shown above. Once they figure out my scheme, they'll know what the price should be for larger primes such as 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, and 61. They can create create their own deeds using index cards and neatly lettering them.

I usually just use the primes from 2 to 11 in the games I run, and here is a nice PDF that creates two sets of professional-looking deeds for those numbers. The PDF is sized to print out cards on standard business card stock, which is pretty easy to get, or you can just print them out on regular paper, cut them out, and attach them to card stock for durability.

One student in each group needs to be the accountant, who will need to keep track of the running totals of money each student accumulates on a scoresheet, so the accountant should start by creating a column labeled with each student's name to track their winnings. Other students need to regularly "audit" his work by watching him to make sure he gets everything recorded properly. (Don't want any budding Bernard Madoffs!) You may want to rotate the job of accountant. Alternatively, each player can keep track of his own earnings--again audited by the other players!

At the beginning of the game, nobody has any money, so there should be a zero at the top of each column. Also, at the beginning of the game, nobody owns any prime numbers, so they should all be placed at the center of the table.

Each student takes turns rolling the two dice. Say he rolls a 5 and 4. Then, he has a choice of declaring his total winnings for that round to be $54 or$45. (You are probably wondering why anyone would choose to go for $45, but you'll see that will sometimes be a useful strategy in later rounds.) Whichever amount he chooses is the amount the accountant records in the books as his winnings for that round. He should keep a running tally, so players know how much money they have. At the beginning of any turn, BEFORE he rolls the dice, if a player has accumulated enough money to buy a prime number that is in the center of the table, then the player may state, "I'd like to buy a prime," and specifies the one he wants to buy. At that point, the accountant subtracts the appropriate amount of money from his total, and the player takes control of that prime, moving it to his place to indicate his ownership. He then rolls the dice as usual on his turn, and records any winnings as usual. From then on, however, when any other player rolls and declares a multiple of his prime, the other player has to SPLIT his winnings with the owner of the prime. Example: Jane has bought the deed for 7. John later rolls a 4 and a 2 on his turn, so he has the choice to declare$42 or $24. If John declares "$42," then John has to split his earnings with Jane because 7 is a factor of 42. Each will get $21 recorded in their columns. On the other hand, if John declares "$24," then he doesn't have to split it--he'll keep the whole $24. More complicated example: Suppose Jane owns the deed for 2 and John owns the deed for 11, and Jimmy rolls a 4 and a 4. Then Jimmy has no choice but to declare 44. Since both 2 and 11 are factors of 44, then it's a three way split. Each player will get 1/3 of the$44. (Generally, I run this game by having the students round off to the nearest whole number, which results in \$15 per student in this case, because it encourages fast mental arithmetic, but it's your decision how to run the game with your students.)

Even more complicated example:

Same facts as above except that assume that Jane is the one to roll the 44. She owns the 2 and John owns the 11. In that case, the split is 1/3 to John and 2/3 to herself.

I generally set a predetermined amount of time for the game to run. At the end of the game, all students can sell their deeds back to the bank for 90% of their original value (good opportunity to talk about asset depreciation!) and the student who has accumulated the most money wins.

After the students have played a game, it's a good idea to discuss strategies. Are some prime numbers better bargains than others? Should you buy the cheapest prime you can afford as soon as you have the money to do so, or should you save up your money and buy a somewhat more upscale prime? This question turns out to be a bit more complicated than people might initially think. Two seems like it would be a great deal (because half of all numbers are divisible by 2), but 3 has special advantages not available to 2.

Some of the same sorts of considerations come up in the game of regular Monopoly. The value of a property depends in part on the amount of rent you can collect when someone lands on your property, but it also depends on the probability that someone will land there. It does little good to own a high rent property if people rarely land there! It turns out that someone named Philip Orbanes has exhaustively analyzed the probabilities of landing on all the different spaces on a Monopoly board. He wrote up his analysis in a great little book called The Monopoly Companion. The book is out of print now, but used copies are available very inexpensively.

Source: This game is loosely adapted from a game called "Pseudo-Monopoly," which I found in the book Family Math years ago, which also had other features, like an income tax, and you can of course adapt this game to incorporate additional features of your choosing as well.