Shapely Pairs
Age 11 to 14 Challenge Level:
Tom worked really hard on these
problems. He used the triangle cards.
Here's what he sent us.
How many possible pairs? I worked this out by picking a card,
working out how many pairs could be made with it using the
remaining cards, and then discarding the card.
The answer is 46, obtained with the following breakdown:
Has all its angles equal: 3 (Does not contain a right angle, Has
all its sides equal, Has three lines of symmetry)
Does not contain a right angle: 9 (Has all its sides equal, All its
angles are of different sizes, Has three lines of symmetry, Has
just 2 equal sides, Has only 1 line of symmetry, All its sides are
of different lengths, Does not contain a right angle and has just 2
equal sides, Has no line of symmetry, Has just 2 equal
angles)
Does not contain a right angle and has just 2 equal sides: 3 (Has
just 2 equal angles, Has just 2 equal sides, Has only 1 line of
symmetry)
Has three lines of symmetry: 1 (Has all its sides equal)
Has all its sides equal: 0
Contains a right angle: 10 (Has just 2 equal angles, Has just 2
equal sides, All its sides are of different lengths, Has no line of
symmetry, All its angles are of different sizes, Contains a right
angle and has just 2 equal angles, Contains a right angle and has
just 2 equal sides, Contains a right angle but does not have a line
of symmetry, Contains a right angle and has all its sides of
different lengths, Has only 1 line of symmetry)
Contains a right angle and has just 2 equal sides: 4 (Has only 1
line of symmetry, Has just 2 equal sides, Has just 2 equal angles,
Contains a right angle and has just 2 equal angles)
Has just 2 equal angles: 3 (Has only 1 line of symmetry, Contains a
right angle and has just 2 equal angles, Has just 2 equal sides)
Has just 2 equal sides: 2 (Has only 1 line of symmetry, Contains a
right angle and has just 2 equal angles)
Contains a right angle and has just 2 equal angles: 1 (Has only 1
line of symmetry)
Has only 1 line of symmetry: 0
Has no line of symmetry: 4 (Contains a right angle but does not
have a line of symmetry, Contains a right angle and has all its
sides of different lengths, All its angles are of different sizes,
All its sides are of different lengths)
Contains a right angle but does not have a line of symmetry: 3
(Contains a right angle and has all its sides of different lengths,
All its angles are of different sizes, All its sides are of
different lengths)
All its sides are of different lengths: 2 (All its angles are of
different sizes, Contains a right angle and has all its sides of
different lengths)
All its angles are of different sizes: 1 (Contains a right angle
and has all its sides of different lengths)
Contains a right angle and has all its sides of different lengths:
0
It is possible to be left with 4 cards at the end, for example:
Has all its angles equal
Does not contain a right angle and has just 2 equal sides
Contains a right angle and has just 2 equal sides
Contains a right angle but does not have a line of symmetry
(and the rest can indeed all be paired).
It is also possible to pair all of the cards. For example:
Has three lines of symmetry, Has all its sides equal
Has all its angles equal, Does not contain a right angle
Contains a right angle but does not have a line of symmetry, Has no
line of symmetry
Has just 2 equal angles, Has just 2 equal sides
Contains a right angle and has just 2 equal angles, Contains a
right angle and has just 2 equal sides
Contains a right angle and has all its sides of different lengths,
All its sides are of different lengths
Contains a right angle, All its angles are of different sizes
Has only 1 line of symmetry, Does not contain a right angle and has
just 2 equal sides.
It is possible to have 6 cards such that any 2 form a pair. For
example:
Contains a right angle
Contains a right angle and has just 2 equal sides
Contains a right angle and has just 2 equal angles
Has only 1 line of symmetry
Has just 2 equal angles
Has just 2 equal sides