1. No category

Cheryl’s Birthday Puzzle
Mathemagic
Explanation & Exhaustive Algorithm
First, I do not understand how there could be just one date, given the
insufficient constraints of the problem.
If Albert is told month M and Bernard day D, then Albert can simply tell
Bernard M , after which Bernard tells Albert D. This is perfectly consistent
with the demands of the problem:
Albert:
“I don’t know when Cheryl’s birthday is, but I know that Bernard
does not know too.”
[so they haven’t told each other yet]
Bernard:
now.”
“At first I don’t know when Cheryl’s birthday is, but I know
[because Albert would have told Bernard M . . . ]
Albert:
“Then I also know when Cheryl’s birthday is.”
[. . . after which Bernard would’ve told Albert D - allowing both to know]
Are we ever explicitly told that Albert and Bernard don’t simply exchange
information in that particular chronological order? No.
Okay, so let us now add some complications - let’s assume they don’t simply
trade information that directly. The expected answer is July 16, but below
1
I go deeper in exposing some of my doubts in one justification I’ve read.
We’re going by the explanation covered in:
http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/apr/13/how-to-solve-albert-bernard-and-cheryls-birthday-maths-problem
My initial answer (without looking up the solution) was June 17. I had
recognized that May 19 and June 18 were untenable on the basis that Albert
was somehow aware of Bernard’s ignorance. But what kind of ignorance?
Suppose Bernard had been told 18 or 19. He could have immediately deduced the months. But does that mean he would have? Without sufficient
intelligence and interest (“intelliterest” or “intellinterest”), he could have
voiced out loud that he can’t tell, on which Albert would have incorrectly
conflated Bernard’s lazy ignorance with true logical impasse. All of this presupposes Cheryl’s honesty as well.
So those are issues.
How did I get June 17, specifically?
Well, before I explain why June 17 is very nearly correct (but still incorrect - it answers a twin problem), we should analyze the claimed solution of
July 16.
2
The linked article states:
All Albert knows is the month, and every month has more than
one possible date, so of course he doesn’t know when her birthday
is. The first part of the sentence is redundant.
The only way that Bernard could know the date with a single
number, however, would be if Cheryl had told him 18 or 19, since
of the ten date options only these numbers appear once, as May
19 and June 18.
For Albert to know that Bernard does not know, Albert must
therefore have been told July or August, since this rules out Bernard
being told 18 or 19.
. . . aaaand there’s where I disagree. Ironically, July 16 does work - but this
justification does not.
Why?
For Albert to know that Bernard does not know, he must simply know that
Bernard was [1] intelligent and interested enough in checking the full implications of what he was told and [2] that he wasn’t told 18 or 19 in order to
cross those out as options.
How would that allow Albert to deduce that the entirety of May and June
get disqualified? Only if he knew Cheryl had told him July or August. But
my question is how do WE know that? We’re just assuming it from the start.
And that’s the flaw with this justification - it presupposes that we know
this. But we don’t.
Ergo, I have developed an algorithm that exhaustively produces all potential
answers, be there one or many. For one, the reason this algorithm is exhaustive is because it literally tests all dates in Cheryl’s chart to see if each
survive scrutiny.
Of course, there are still implicit assumptions that need to be told, but
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these are assumptions that most likely correlate with the original intentions
of the testmaker. They are:
[] Cheryl is completely honest.
[] Cheryl doesn’t just blurt out the answer so both can hear.
[] Albert is told the month in secret, and Bernard is told the day.
[] Albert and Bernard do not communicate these results candidly (otherwise
we wouldn’t have a puzzle).
[] Albert and Bernard can give each other minimal hints or the eventual answer only with maximal delay of hint-giving - all in order to reach an answer.
[] Albert and Bernard are intelligent and interested and pursue the implications of every hint they exchange.
Fair enough? Okay, let’s proceed.
So my algorithm is basically relentlessly analytic. It assumes every single date in the chart is told, and keeps going until ambiguous impasses are
reached. It’s really that simple. Dead-end? No-go. No dead-end? Possible
solution.
This way, unlike the article, we are not pretending to be oracle machines
that have access to presupposed knowledge we don’t actually have. In other
words, we are not telling the reader that it’s July 16 already - we are simply
entertaining the idea that each date was given and seeing whether they add
up with the claims Albert and Bernard make in the latter half of the puzzle.
The possible dates are:
May 15, May 16, May 19, June 17, June 18, July 14, July 16, August 14,
August 15, August 17
We begin.
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[] Assume Cheryl tells them (May,15)
[] Albert knows (May)(15?,16?,19?)
[] Bernard knows (May?,Aug?)(15)
[] Albert disqualifies (18,19) due to Bernard’s confession of being unable to
deduce Cheryl’s birthday using just his information.
[] Albert now knows (May)(15?,16?)
[] Bernard knows (May?,Aug?)(15)
[] We still have ambiguities. Impasse.
[]
[]
[]
[]
[]
[]
[]
Assume Cheryl tells them (May,16)
Albert knows (May)(15?,16?,19?)
Bernard knows (May?,Aug?)(16)
Albert disqualifies singlets (18,19)
Albert now knows (May)(15?,16?)
Bernard still knows (May?,Aug?)(16)
We still have ambiguities. Impasse.
[] Assume Cheryl tells them (May,19)
[] Nonsense. Bernard’s puzzle-implied intelligence/interest would have allowed him to know the month immediately, but he claims ignorance. Impasse.
[] Assume Cheryl tells them (Jun,17) (this was the solution I had believed at
first)
[] Albert knows (Jun)(17?,18?)
[] Bernard knows (Jun?,Aug?)(17)
[] Albert disqualifies singlets (18,19)
[] Albert now knows (Jun)(17)
[] Bernard still knows (Jun?,Aug?)(17)
[] Albert has reached a singlet, which he can then communicate to Bernard.
But June 17 couldn’t be valid for this particular problem, since we are told
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that Bernard learns the answer before Albert. Had we been told Albert
learned it before Bernard, it would have been one valid answer.
[] Assume Cheryl tells them (Jun,18)
[] Nonsense. Bernard’s puzzle-implied intelligence/interest would have allowed him to know the month immediately, but he claims ignorance. Impasse.
[]
[]
[]
[]
[]
[]
[]
Assume Cheryl tells them (Jul,14)
Albert knows (Jul)(14?,16?)
Bernard knows (Jul?,Aug?)(14)
Albert disqualifies singlets (18,19)
Albert still knows (Jul)(14?,16?)
Bernard still knows (Jul?,Aug?)(14)
We still have ambiguities. Impasse.
[] Assume Cheryl tells them (Jul,16)
[] Albert knows (Jul)(14?,16?)
[] Bernard knows (May?,Jul?)(16)
[] Albert disqualifies singlets (18,19). But furthermore, he disqualifies all of
(May,Jun,Aug) since he knows it’s Jul already.
[] If Albert is merciful and communicates the hint to Bernard that (May,Jun)
don’t apply, Bernard then knows (H
May,Jul)(16).
H
H
Z
[] If Bernard gives Albert the hint that it isn’t 14, then Albert knows (Jul)(
14,16).
Z
[] Both have identified July 16 without giving each other the answer directly.
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[]
[]
[]
[]
[]
[]
[]
Assume Cheryl tells them (Aug,14)
Albert knows (Aug)(14?,15?,17?)
Bernard knows (Jul?,Aug?)(14)
Albert disqualifies singlets (18,19).
Albert still knows (Aug)(14?,15?,17?)
Bernard still knows (Jul?,Aug?)(14)
We still have ambiguities. Impasse.
[]
[]
[]
[]
[]
[]
[]
Assume Cheryl tells them (Aug,15)
Albert knows (Aug)(14?,15?,17?)
Bernard knows (May?,Aug?)(15)
Albert disqualifies singlets (18,19).
Albert still knows (Aug)(14?,15?,17?)
Bernard still knows (May?,Aug?)(15)
We still have ambiguities. Impasse.
[]
[]
[]
[]
[]
[]
[]
Assume Cheryl tells them (Aug,17)
Albert knows (Aug)(14?,15?,17?)
Bernard knows (Jun?,Aug?)(17)
Albert disqualifies singlets (18,19).
Albert still knows (Aug)(14?,15?,17?)
Bernard still knows (Jun?,Aug?)(17)
We still have ambiguities. Impasse.
All dates have been checked. As far as singlets go, only June 17 and July
16 are plausible. The problem’s chronological order stipulates that July 16
be the victor.
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In conclusion, it really depends on the strictures applied. Remove too
many, and literally every date becomes a possibility, as was proven in the
start.
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