A circle is inscribed in an equilateral triangle. Smaller circles
touch it and the sides of the triangle, the process continuing
indefinitely. What is the sum of the areas of all the circles?
Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP
have equal areas. Prove X and Y divide the sides of PQRS in the
A 1 metre cube has one face on the ground and one face against a
wall. A 4 metre ladder leans against the wall and just touches the
cube. How high is the top of the ladder above the ground?
It was nice to see several younger
solvers dipping their toes into this introduction to group theory!
Whilst we have some nice thoughts so far, we feel that there is
more milage in this problem, so we are leaving it open as a tough
nut for the more advanced students to consider.
Shriram from Canberra Grammar in Australia
noted that you win the game if you save the choice of antidotes
until the end of the game, although it would be luck if this
happened. However, this shows that the order does matter! Shriram
felt that you would lose the game 33% of the time through
dxm from Wilsons primary school
attempted a nice systematic approach by noting that if you start on
the left then you die most of the time.
Daniel from Savile Park Primary sent us a
lovely set of thoughts, which is very impressive when his age is
taken into account! We liked the fact that he created a simpler
version of the problem to get a feel for the ideas and were
delighted with his version of the game in which the order does not
matter! His solution is as follows:
Does the order matter?
For the case that there are only 3 cards and the middle one is
turned over (if it matters for this; it will matter in general, and
3 is easier than 5):
Live or Die
If 2As and 1P or vice versa it matters. But only 2P 1A changes
Chance of being poisoned
From table chance of death=?
All 27 are equally likely starting from 5 cards because:
Scissors, Paper, Stone
The order sometimes matters.
If the 3 cards are all different then the order matters, but the
middle one can't win.
e.g. Sc, P, St can be won by scissors or stone
Cards with 1,2,3 on, largest number wins.
Order doesn't matter.
Well done Daniel!
Other solvers noted that although the game as
it stands is random, for a PARTICULAR set of cards sometimes you
live and sometimes you die. One unnamed solver noted that sometimes
the game has a guaranteed outcome, for example if all of the cards
are the same, which occurs with probability 1/81, and enumerated
all other possibilities in a spreadsheet, suggesting that the
chance of survival is 1/3 for the Poison, Antidote, Water game.
Another unnamed solver created a nice
formal algebraic notation to aid with the analysis, although this
is a 'formal' representation, as adding of + infinity and -
infinity is undefined in arithmetic:
If you mix antidote and poison together you end up with water.
If you mix either with water nothing happens.
But if you mix one substance with the same substance nothing happens.
Therefore the effects must be like the list below.
Poison = -i (standing for infinity)
Antidote = +i (standing for infinity)
Water = 0
Taking one of my example cases (involving 2 poisons, 1 water and 2
-i-i = -i
-i+0 = -i
i-i = 0
0+i = i
But if you rearranged the order in which this occurred:
i+i = i
i+0 = i
i-i = -i
-i-i = -i
I would have died
Taking another circumstance (involving 3 poisons and 2 waters):
0-i = -i
-i-i = -i
-i-i = -i
0-i = -i
I died and there was nothing I could do about it.
The order sometimes matters, but not always.
If anyone has any other thoughts concerning
this problem do let us know; solutions will be left open!