2020 Mathematics Specialist 11: Investigation 1 (Version 2)
Take-home part – Syllabus Topic 1.1 (points 1.1.1-1.1.9)
Robot O and Robot X are bored. To pass the time they decide to invent a game using a row with 6
boxes.
O X X O X O
They take it in turns to add their own symbol randomly to one of the empty spaces on the grid until
all the spaces are filled. At the end of each game, there are always three Os and three Xs on the grid
e.g.
O X X O X O
1. Calculate the total number of distinct completed grids.
In an attempt to add some excitement to the game, Robot O suggests a scoring system, where each
row of 3 identical symbols in a completed grid earns 1 point for that player. (Note that the order in
which symbols were added is irrelevant; i.e. it doesn’t matter which robot created a row of 3
identical symbols first.)
E.g.
O X X
X
O O
earns 1 point for Robot X and 0 points for Robot O.
2. Calculate the following probabilities:
a) Robot X earns 1 point and Robot O earns 0 points at the end of a game.
b) Robot X and Robot O each earn 1 point at the end of a game.
c) Robot X and Robot O each earn 0 points at the end of a game.
Robot O now suggests a new game, to be played on a 3x3 grid. Once again, the robots take turns to
add their own symbol randomly to one of the empty spaces on the grid. The middle square is left
empty so that there are always four Xs and four Os in a completed grid, e.g.
O X X
X
O
O O X
At the end of each game, points are awarded as before, so that each row of 3 identical symbols
(horizontal or vertical) in a completed grid earns 1 point for that player.
E.g.
X X O
X
O
scores 1 point for Robot O and 0 points for Robot X.
O X O
X O O
X
O
X X O
scores 1 point for Robot O and 1 point for Robot X.
3. Investigate how to calculate probabilities for different outcomes (e.g. the probability that
both robots score 0 after a game, or the probability that Robot O scores 1 point, etc.)
In the middle of a thrilling game, the two robots notice that their friend, Robot Y, has arrived.
“Can I join in?” it asks them. “Sure!” reply Robots X and O in unison. They adapt their games so that
Robot Y can participate. To help Robot Y learn how to play, they begin with the simpler game, this
time using a row of 9 boxes:
O X X
X
O O
The three robots take turns to randomly add their symbol to an empty box, until the grid is
complete, e.g.
O Y
Y
X
O O
Y
X
X
As before, they use a scoring system where each row of 3 identical symbols in a completed grid
earns a point for that robot.
4. Investigate how to calculate probabilities for different outcomes (e.g. the probability that all
3 robots score 1 point after a game, or the probability that Robots X and Y both score a
point, but Robot O does not).
After a few hundred million games, Robot Y asks “Could we try to play on a 3x3 grid now?”
“Certainly!” says Robot X. “And I guess we should include the middle square now, so that there are 9
squares in total – that way there’ll be exactly 3 of each symbol in a completed grid.”
“So now there’ll be diagonal rows as well – this sounds like fun!” says Robot Y.
“Yeah…” says Robot O, with just a hint of wistfulness.
“Something wrong?” says Robot X.
“Well… it’s nothing really… I’m just so BORED with horizontal and vertical rows! I really wish we
could play with just diagonal rows!”
“We could,” says Robot X, “but that wouldn’t be very interesting, as there are only two different
ways to make a diagonal row of 3 symbols in a 3x3 grid.”
Robot Y is momentarily lost in thought. Eventually it says: “What if we generalise the idea of a
diagonal row?”
“What do you mean?” chirped Robots O and X simultaneously.
“Well, let’s say I make a diagonal row of Ys in a 3x3 grid. A nice property of this arrangement of Ys is
that no two Ys are ever in the same horizontal or vertical row.”
Y O X
X Y X
O O Y
“Go on,” say O and X.
“What if we made that the defining property for a configuration that scores a point?” says Robot Y.
“In other words, each of us scores a point after a game if and only if no two of our own symbols are
in the same horizontal or vertical row. Here, I’ll show you some other examples of ways I could score
a point.”
X O Y
X Y X
Y O O
Y O O
X X Y
X Y O
“So in the last one, all 3 of us score a point – looks interesting!” says Robot X.
“I agree!” says Robot O.
“Then let’s give it a try!” says Robot Y.
To be continued…
O Y X
X O Y
Y X O