The Birthday Problem

Several people have queried me on this and thus here’s a detailed explanation.

This is actually an ancient Math logic problem (with many versions).

The idea to solve it “logically” is like this:

(1) A person only “doesn’t know” if there are at least “two possible answers” remaining from his “point of view”.

(2) A person “knows” the “answer” if there’s exactly only “one – unique” possible answer remaining.

In our case here, the “answer” is a “birth-date”.

Also, Albert only “knows” the “month” while Bernard “knows” the “date” (because Cheryl told this to them separately).

Let’s begin to solve it now.

Step #1: Albert first says that he doesn’t know the birthday.

So, knowing the month, two avoid situation (2) above (which will make Albert know the answer instead), the answer must be a “month with no unique dates” since “Albert also says that Bernard does not know too from the initial point of view.”

From our list,

Months with unique dates are only May or June since the numbers 18 & 19 only occur “once” (unique). The months they occur in, namely “June” (18) and “May” (19) are thus “rejected” to satisfy Albert/Bernard’s not knowing.

Hence, the two remaining months “July” or “August” are the possible months now —> (3).

Step #2: Bernard says that he knows now!

Bernard, by the turn of possibilities at the end of step #1 above, realizes the answer now.

Bernard knows (3) being a logical thinker.

Look at the possible dates in July and August.

The problem states that at this stage, he “knows” the “answer”. Hence situation (2) “must” happen and thus “any possibility” which amounts to situation (1) must thus be “rejected”.

From (3), we know that the “month” is either “July” or “August” (the latest information to date).

We “reject” the date “14” because situation (1) arises since there are “two” possible answers for this date, namely, “July 14” and “August 14”.

At this stage, the “remaining” possibilities are:

July 16, August 15 or August 17.

To “know” the answer, situation (2) must arise. Since the “dates” are all “different” here, we now look at the “months”.

From the list of the three remaining possibilities above, we find that the month “August” still gives rise to situation (1) since there are “two” dates here (15 or 17) with this “same month” (August).

Hence, only “July 16” yields situation (2) of a “unique-one” answer at this last step to cause both Bernard & Albert to know Cheryl’s birthday.

P/S:

A common remark, “solving a problem is the science of Mathematics but presenting your solution is the art of Mathematics”.

P/S 2:
Here’s a little clarification which may help edify the argument above.

Many people argue that the ‘rejection’ of the months May and June is ‘not logical’ in step #1.

To understand ‘why’ it’s such, we must understand the meaning of “not knowing” as it’s implied in ‘logic’ (not necessarily as in English).

Simply put, ‘not knowing’ in Logic is meant in the “absolute sense” while in English, it’s either in the “absolute or relative sense”.

So, in our given example (focusing on step #1),

(1) For Bernard (who was told the ‘day’) to “know” the birthday “absolutely” at the ‘beginning’ itself, the only way is for the date to be ‘unique’.

Since this is ‘not’ the case (given that – Bernard does “not know” instead), thus months with “unique dates” must be “rejected” to satisfy a ‘not knowing’ in the absolute sense with respect to him.

2) Also, Albert (who was informed about the month) also does “not know” at step #1.

Thus to satisfy an “absolute not knowing” for him, we must “reject” any months which may have “unique characteristics”.

For example, a month with only “one date listed” can be considered unique too but it’s not the case here.

The only ‘unique characteristic’ that exists among the months from the “beginning” point of view (looking at the whole set) is that there are exactly “two months” (May & June) with “unique” dates (19 & 18 – occurring “once”) respectively.

Thus, since we speak ‘absolutely’ in logic, to have the “absolute” situation of “not knowing” at all for Albert here (Probability of Albert ‘not knowing’ is to be One), it must be that the months are “not May/June” causing them to be “rejected” in step #1.

If the argument was by English language, a ‘relative’ statement holds and it remains inconclusive at step #1.

Why?
In English, “not knowing” does not necessarily mean an “absolute no” (as it is in the field of ‘logic’) but also a ‘maybe’ (due to a possibility of either a ‘yes’ or a ‘no’ to occur resulting in a state of ‘not knowing’ too).

In our case (at step #1), English-wise speaking, from Albert’s (given – month) point of view, May/June is still possible because when he says “don’t know” in the “beginning” it may mean:

With respect to months May/June, we have the three possible scenarios below:

(a) A possibility for an ‘absolute no’ – ‘absolute not knowing’ (reject May/June)

By rejecting the months May/June, we have an “absolute case of not knowing” (with respect to the ‘beginning’ and from their respective points of views ‘only’) for both Albert/Bernard since the “remaining” months July/August each do “not” have “any unique” characteristics when viewed with respect to “all” months (May/June/July and August) at the ‘start’.

Thus for this case to happen indefinitely, Albert must have heard a different month told directly to him.

(b) A possibility for an ‘absolute yes’ – for an ‘absolute knowing’ (accept a unique date only)

He may have heard a statement implying a “unique month” –> months with ‘any unique’ characteristics –> in this case, only months with a unique date-number, namely, 19 or 18 (Thus if he heard the month May and knew ‘somehow’ it’s unique, he would ‘know’ that the answer is May 19. Similarly, if he heard the month June and knew ‘somehow’ it’s unique, he would ‘know’ that the answer is June 18).

(c) A possibility for ‘relative answer, yes or no’ – “relatively not knowing” because it’s either a ‘yes’ or a ‘no’ (it may still be May/June)

When Albert was just told the month, say May/June, he still doesn’t know because it could be a

(i) Possibility of a ‘yes’ –> (for unique dates only possessed in these months May 19 or June 18)

(ii) Possibility of a ‘no’ –> (that is we can’t know since the other dates listed here are inconclusive “not” being unique in any way).

–> which does “not” result in an “absolute” state of “not knowing” as “logic requires” (that is, he doesn’t know because suppose he heard ‘May’, if this was the case, then each of the dates May 15, 16 or 19 is a ‘yes’ or a ‘no’)

Thus we must reject (c) due to it not yielding an “absolute not knowing”. We reject (b) because it yields an “absolute knowing” in the “beginning” itself which is “not” what the problem describes.

The rejection of (b) and (c) yields the only possibility of accepting (a) to get a ‘situation’ of “absolutely not knowing” in the “beginning” itself as required/stated in the problem statement.