素数遊 The Japanese characters mean "Prime Number Play"

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fig. 1. So-Soo-Yoo board, pieces, dice, rulebook |

#### Background

I first started thinking about the design for the game about twenty years ago when I was playing

*Chute and Ladders*with my son when he was small.fig. 2. Chutes & Ladders Game |

The game is simple, easy to learn and to play, but it occurred to me that if one were to learn, to memorize, all of the connections between the one hundred spaces on the board, one would not be learning anything

*valuable*about mathematics. I began to ask myself if it would be possible to make a game with a similar form and style of play to*Chutes & Ladders*which*did*teach valuable information that would be reënforce these skills the more one played.I also was eager to spare my son having to memorize the multiplication tables (or "ku-ku" 九九 as they call them in Japan--"nine by nine"). It occurred to me that the multiplication tables would be a useful thing on which to base the game. It then hit me that prime factors as a basis for being able to jump around the board, in other words, primes and combinations of primes as the "chutes" and "ladders" of the game I was hoping to design.

#### Development of the Game

Dr. David Chappell's excellent "scaffolding" diagram elegantly illustrates what I was wrestling with in trying to design the board and envisage the mode of play. The idea was that every legal move would be a jump between some two multiplication tables. Two spots that have one or more prime factors in common are connected, that is, they have a "chute" or a "ladder" between them.

fig. 3. Jumping between multiplication tables by common factors |

Obviously the number connections, the degree of interconnectedness, between the spaces on the board was high, much higher than with

*Chutes & Ladders*. I had to figure out how to represent this on a board such that it would not be too complex and would actually be playable as a game. It would also have to be easy to learn, including by small children.My first idea was a "map" of "countries" possibly with "rivers," each with a number, but this of course broke down quickly. Thirty is the first number with more than two prime factors, 2 x 3 x 5, so "two land" and "the three forest" and "five land" would all have to meet at something like "Thirty City" and so on.

I initially imagined "freeways" numbered for each prime number, e.g., the "Number Two Freeway" or the "Number 13 Freeway," going round and round from a center at the number one, each new freeway starting when it's number came up, with "exit ramps" for each number that shared the same prime factors. In David Chappell's figure 3 the vertical rainbow colored lines are the same sort of thing I was considering with my "freeway" idea. The number of freeways would be high, so using colors to distinguish them was a problem.

My original designs started to look like a Mayan calendar, and the size and complexity started to look unmanageable, and how the game might be played was still unclear.

fig. 4. Mayan Calendar |

I then hit upon the idea of simply removing the "freeways" and making the connections between the spaces on the boards based on factors only, with no explicit "pathway" drawn between the board spaces. This gave me the board design as it stands now.

Here, rather than being represented as a "freeway" giving access to other spaces, the prime numbers are simply the spaces with "only one prime factor." The spaces for non-prime numbers have collections of miniature versions of the prime spaces in the form of "bubbles" that show all of their prime factors. I wanted to use colors on the board, but the number of prime factors was too high for a six-color system. I hit on the idea of having a system of patterns that would change every six prime numbers, starting with solid color, then a ring boundary, then a single stripe, then a cross, and finally a central dot (on space number 97). Since 97 is the last prime number that appears as a prime factor on the 200-space board, all prime numbers after that are just plain white.

Now I had the board designed, and I had to work out how the game was actually going to be played.

#### Mode of Play: Dice

I arbitrarily picked a six-sided die to determine how many moves could be made per turn. A six-sided (Las Vegas) die is familiar to everyone. I had to think about the other dice, however.

It was a problem to determine how to place the tokens and the player pieces such that players would be forced to move around the board, preferably with randomly-determined goals. The goal was of course to force players to prowl around this "map" and thereby learn all of the lessons to be learned over time. These lessons included the multiplication tables, factors, powers and so on.

That's why the board goes up to 199, by the way. I had initially made it to go up to 149, based on the American school practice of studying up to the twelve multiplication tables. This required modifying a twenty-sided die (d20), and I later decided it wasn't worth it.

The board was kind of big, but in fairness only twice the size of

*Chutes & Ladders,*which is played by nursery school children. I wanted to come up with a system for placing pieces randomly on the board. I thought about things like cards, dreidels, a double- or triple- layered spinners. I finally came up with a d20 and a d10, read in order, to give a value between one and two hundred, with two hundred (or zero) being a penalty roll, causing loss of a piece. If the d20 has a twenty and no zero (which they all seem to these days) then a twenty is treated like a zero.In other words, the die are read from the d20 to the d10, not added, so it's easy for small children. Further, this is easy since reading "17 and 3" obviously matches the 173 space, and small children simply have to compare the symbols to find the space from the dice, without having to do any computations, or even understand how big the numbers are (although this will be learned over time).

So there it is. That's how the dice work.

#### Mode of Play: Player Pieces

The initial idea was just that there would be tokens, like coins, and player pieces would just be miscellaneous trinkets, like in classic

*Monopoly*.Initially there was no fixed requirement as to how many player pieces were needed, though in keeping with the whole six-color scheme, I finally went with six player pieces, one of each color with matching tokens.

Another arbitrary choice was that to win you need to collect six tokens (or "rings" in the current implementation). Since there were six colors in play, I decided six tokens were needed to win, and again arbitrarily, that one token of each color or six of one color were needed.

Those are the design decisions that led to the shape of the game as it now stands. At this point, it's good to look at the rule book to see how the game is played.

#### How Well Does it Work?

The answer would be "very well." Small children, and even adults, can learn to play skillfully very quickly. I've tested it on children as young as nursery school, and I have played it with children in French, English, and Japanese. All are able to learn and begin to play quickly.

It's an "abstract game" which means that it does not require outside knowledge of things like accounting, math, geography, etc., in order to play, or such knowledge would give an advantage to a player possessing such skills. As such, a nursery schooler may play on even footing with an engineer or retired math professor, and this fact has already been demonstrated. Indeed, since small children tend to just play the rules, as opposed to trying to "figure it out," they tend to make moves more quickly, and even make better moves, than many engineers would. In other words, the game is not "difficult" either to learn or to play, but there is a lot of latitude to "overthink things."

#### Do Kids Actually Learn Math?

I have done a lot of testing, but this is still something of a question mark. As I've stated, a child does not

*need*math skills to play the game, and indeed that*having*math skills (or a PhD in mathematics) does not seem to make people "better" at playing the game.One is passively and repeatedly exposed to the multiplication tables by playing. Ask a child who has played a lot of So-Soo-Yoo if 42 or 49 are factors of seven, or to recite the six multiplication tables, while these

*would*be things that the child has been exposed to by the game, but would the child have learned this? We don't have a lot of data on this yet.Would they know the difference between 54 and 56? Children the world over have trouble keeping 9 x 6 = 54 and 7 x 8 = 56 straight, since they're just memorizing. In So-Soo-Yoo these two spaces are next to one another, just like half the spaces on the board, but strategically they are totally different. In principle a child who learned the multiplication tables from So-Soo-Yoo would not confuse 54 and 56.

That's the idea--that kids play the game over and over with each other, pushing each other to get better, checking each other's adherence to legal moves, and chasing each over all over the "map" of the natural numbers, hopefully learning all their details and secrets. Any math fact they learn would be some correct move they made, perhaps helping them to win or "bump" an opponent, which might make it memorable. There are "highways" and "hotspots" in the board, or on the number line, which have more factors, or "deserts" which have few and are hard to reach, and experiencing these in the game may drive them home.

Oh, yes, and obviously this game may be played by parents, children, and siblings, and it's just as challenging for all ages. This can offer an opportunity for parents or older siblings to illuminate math facts and principles during play, as they happen.

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