Triplets
Master Euclidean triangles with this exciting game of wits and strategy!
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DECODING THE TRIANGLE
Maybe the most famous of all mathematical theorems, The Pythagorean Theorem is pretty wimpy as far as lead singers go. If you need a reminder, click Start for a visual proof.
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The idea that a²+b²=c², only works on a flat, 2D, right triangle. That's like saying a breakthrough medicine only works on 43 year-old, bald, male, PE teachers. How can it possibly be that wonderful if it is only valid for such a small group of people? Similarly, the Pythagorean Theorem is only useful for a tiny group of triangles. However, with some ingenuity and logic we can expand this famous theorem into something much more helpful.
A²+B²=C²
Starting with two sides (A and B), we built 3 different triangles. In the middle, we made a right triangle (the bald PE teacher). On the left, we made the angle between A and B smaller than a right angle. On the right, we made the angle between A and B bigger than a right angle.
With some clever geometry (you'll need to read The Shape of Everything to find out more), we know the sides are related to their opposite angles so that D<C<E.
Since our side lengths are all positive, that means D²<C²<E².
A little substitution and we find D²<A²+B²<E².
Why is this cool? We just gave that old dusty theorem a power up! The Pythagorean Theorem can now tell us if an angle in a flat, 2D triangle is acute, obtuse, or right.
Acute, Obtuse, or Right?
Acute
If the side squared is smaller, the opposite angle is acute.
D² is smaller than A²+B², so the angle formed by sides A and B is acute.
A²+B²>D²
A²+B²=C²
A²+B²=C²
A²+B²<E²
Obtuse
If the side squared is larger, the opposite angle is obtuse.
E² is larger than A²+B², so the angle formed by sides A and B is obtuse.
Right
If the side squared is equal, the opposite angle is right.
We proved in a flat, right, 2D triangle A²+B²=C².
A²+B²=C²
Is it a Triangle?
With our Pythagorean Power Up, we have a way to determine by the side lengths if an angle in a triangle is acute, obtuse, or right.
But will any three side lengths create a triangle? Try out the activity below to unlock this mystery.
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The Triangle Inequality
If a triangle has sides of length a and b, the third side, c, must have length less than the sum of a and b and more than the positive difference of a and b.
Umm. I don't get it.
Imagine you are going home from school or work.
Okay...
You get hungry and decide to grab a bite to eat.
As I often do...
Your trip must be longer with this detour.
a
b
c
Adding up your milage, a+b >c.
Going out of your way creates a triangle.
It will always be longer to make an extra stop than if you just went straight home.
Unless the restaurant is on the way!
Then you don't have a triangle!
a+b=c
a
b
c
Object:
Place cards to score points. The player with the most points is declared the victor!
Getting Started:
The game begins with a Joker placed. The Joker acts as a wild card. The first player places two cards adjacent (next to) the Joker either vertically or horizontally.
Play:
Play continues with each player taking a turn by placing two of their cards on the board. Players must place their cards off a previously played card (either vertically or horizontally) to create a run. Each run can only contain three cards. This means you cannot place a card such that it is touching another run or any other card not included in your own run. The game ends after a total of 13 rounds, allowing each player to place all 26 of their cards.
How To Play
Scoring:
Each card represents a side length. In forming a run, players create a triangle that may be acute, obtuse, or right. The order in which they place their side lengths (read left-to-right or top-to-bottom) determines which angle of their triangle they wish to play. Players earn points based on the angle. (See Acute, Obtuse, Right for how to determine the type of triangle and angle given three side lengths.)
Acute Angle: -2 points
Obtuse Angle: +2 points
Right Angle: Triples your current score
That's Impossible:
If a player sets down a run that cannot form a triangle at all, their score is reset to 0.
Example:
It is Player 1's turn and they have the cards: 5, 7, 12, 17, and 21. They want to play off the 13. They remember sides with length 5, 12, 13 create a right triangle and hope to triple their score.
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Run Error:
They wish to play their 5 and 12 to the left of the existing 13. This is an invalid play as the 5 would intersect with the existing run, 2-6-15.
Order Matters:
They consider playing to the right of the 13 instead. However, playing 13-5-12 or 13-12-5 plays the acute angle of their right triangle. This is a valid play, but they would lose 2 points.
Impossible Triangle:
Since they must play to the right of the 13, they decide to play a smaller card to help build an obtuse angle. Placing their 7, they consider adding 21 to complete their run. They figure 21² must be larger than 13² +7². While this is true, a 13-7-21 run is an impossible triangle. That is, given side lengths of 13, 7, and 21, a triangle cannot be formed. Playing this would cause the player to lose all their points, resetting their score back to 0.
The Play:
Player 1 ultimately decides to create a 13-7-17 run. As 13²+7²=218 and 17²=289, they have 218<289 and have played an obtuse angle, gaining 2 points.
Tips:
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Tripling a large score has a bigger impact - consider saving key cards till later in the game.
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Consider what cards can be played with your 1 to avoid an impossible triangle and plan ahead.
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Think about what runs (groups of 3 cards) build a Pythagorean Triple and play these cards carefully.
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