Finding Common Ground: Common Features in Apparently Unrelated Games
I posted a single blog with the same title several years ago, but this series has a broader focus than solving math puzzles.
What is a Take-away Game? Tic-Tac-Toe is an example of a take-away game: two players take turns placing X’s and O’s on open squares of a 3×3 grid. The objective is to be the first to fill a row, column or diagonal with three of your marks. What is “taken away” by each move is an open square.
In England the same game is called “Naughts and Crosses”, which is a better descriptor than the American name. A quick search reveals an interesting hypothesis about the origin of the term, “Tic-Tac-Toe”.
With best play by the two opponents, games of Tic-Tac-Toe on a 3×3 grid end without a winner. By contrast, on a 2×2 grid the first mover has a trivial win. You can play Tic-Tac-Toe on bigger squares, rectangles or 3-D objects.
Games are wonderful teaching devices for three reasons: (1) games reward strategic thinking; (2) games often share an underlying structure, despite different outward appearances; (3) games are fun, especially with friends or family. Spontaneity may be a problem in a classroom setting, but it works well at home.
In short, games can both educate and entertain.
Disclosure: My blogs on takeaway games are not lectures in Game Theory, a branch of applied mathematics with broad applications in all of the Social Sciences. They do present ideas you can understand and games you can master using only Grammar School arithmetic. Like Simple Division: 38 divided by 12 equals 3 remainder 2. You can form at most 3 groups of size 12 from 38 items, leaving 2 ungrouped items.
Full disclosure: These blogs are inspired by a question I asked myself during the summer of 1950, a question that lay dormant in my mind for nearly 50 years. Back then, I had no idea I was seeking winning strategies for a take-away game played on an amusement park ride. I won’t tell you the ride that inspired my interest in take-away games but you’ve probably guessed it anyway. And I can’t bring that full experience to your homes, but I can offer a set of games that are variations on the original theme. More on those, soon.
In the summer of 1950, I was also unaware that two childhood activities provided clues to answering my question: Fortune-telling with Daisy Petals and chanting nonsense rhymes like Eeny, Meeny, Miney, Moe. This opening blog is titled “Finding Common Ground”, because it identifies a common feature in those two apparently unrelated games.
Fortune Telling with Daisies
This practice supposedly originated in 17th century France. A person wants to learn if his/her amorous feelings are reciprocated, and selects a daisy to find out the true state of the romance. The questioner plucks daisy petals while alternating two statements: “My love loves me”; “My love loves me not”. The statement associated with the last daisy petal is supposed to be the correct one.
Clearly, the questioner wants this procedure to yield a positive outcome. Is there a way to guarantee that the questioner says “My love loves me” while plucking the last petal?
Notice that Fortune-telling With Daisy Petals is not a “game” in the usual sense: there is no active opponent. You can pluck petals in any order, but you must alternate the two statements. The questioner is seeking a playing strategy without having to worry about a reaction from the daisy.
Eeny, Meeny, Miney, Moe
This is also a “game” with no active opponent. Children all over the world have created nonsense rhymes to choose Who’s “It”. Sometimes it is good to be selected: that child might be able to pick a friend to be a team member. Being selected is unfortunate when there is one too many for a fun activity; the child associated with the last word of the nonsense rhyme has to sit out the next play session.
If you are the “chant-caller”, can you arrange for a friend (or yourself) to receive the positive outcome, or for a rival to receive the negative one?
Fortune-telling With Daisy Petals and choosing Who’s “It” with Eeny, Meeny, Miney, Moe are examples of take-away games: daisy petals are plucked and silly words assigned until none is left, at which point the game ends. That is the common feature in these apparently different activities.
Surprisingly, marking the passage of hours on an analogue clock provides an insight into both of these games.
Blog #2, “Clock Arithmetic”, resembles a lesson plan your 4th or 5th grade teacher might have used. Revisiting that class is worthwhile, even though digital displays may now outnumber clock faces. On a lighter note, I will show how Clock Arithmetic can be used in parlor tricks that make you appear to be a calculation prodigy.
Again, full disclosure: I don’t know any riddles or puzzles that Clock Arithmetic helps you solve. Furthermore, Clock Arithmetic only provides insights into a certain class of Take-Away games. So I was surprised to discover that Clock Arithmetic is relevant to the question I asked myself seventy-two years ago.
Here are a few things you might like to try:
- Let loose your search engine on the origin of the name, “Tic Tac Toe”.
- Think about “winning” strategies for Fortune-telling with Daisy Petals and Eeenie, Meeny, Miney, Moe
- Let me know your favorite childhood nonsense rhyme. If you grew up in another country, you probably have something analogous to Eeny, Meeny Miney, Moe.
“So, what is that amusement park ride”, you might ask? You can probably guess it but please bear with me, because I intend to share a lot of neat stuff in upcoming blogs.
Among other things, I will demonstrate a simple apparatus for playing the take-away game that inspired my question 72 years ago. And I’ll introduce some pretty neat variations on that original game.
Games also keep your mind active and sharp. Participating reaps benefits far surpassing passive viewing of others playing games. My childhood game-playing, circa 1950, was inventing baseball games using dice, employing the names of real players on real teams. Fantasy baseball, perhaps, in today’s world.
A non violent game. I’m looking forward to playing it.
Interesting read and thoughtful insight. Thanks!