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Poison, Antidote, Water

Stage: 4 and 5 Challenge Level: Challenge Level:1

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 luck.

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):

Cards

Outcome

Live or Die

PPP

P

D

PWP

P

D

PAP

P

D

APA

A

L

WPW

P

D

WWW

W

L

AWA

A

L

AAA

A

L

WAW

A

L

APP

P/W

D/L

PPA

W/P

L/D

WPP

P

D

PPW

P

D

AWW

A

L

WWA

A

L

PWW

P

D

WWP

P

D

PAA

A/W

L

AAP

W/A

L

WAA

A

L

AAW

A

L

PAW

W

L

APW

W

L

AWP

W

L

PWA

W

L

WPA

W

L

WAP

W

L

 

If 2As and 1P or vice versa it matters. But only 2P 1A changes survival.

Chance of being poisoned

From table chance of death=?

All 27 are equally likely starting from 5 cards because:

PP-> P

WP-> P

WW-> W

AP-> W

AW-> A

AA-> A

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

Last Part

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
antidotes):
-i-i = -i
-i+0 = -i
i-i = 0
0+i = i
I survived
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!