William M. Briggs

Statistician to the Stars!

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The Logic Puzzle That Is “Stumping The Internet.” Update: Now With Solution


The Washington Post thinks the puzzle above is hideously difficult and says that it has become an “obsession”. The writer there certainly finds it sweat-inducing: “As the problem spread far and wide, it became a bit of a joke. Truthfully, we blame no one for choosing to laugh instead of crying out of frustration.”

Problem originates from Singapore, hence the dicey English. “The question is actually from a test given to students who are sophomores or juniors in high school”. Word has it that parents are “angry” over the question. I’m dubious. But somebody did mistakenly say that the puzzle was for younger kids. And you know how people like to check facts before commenting.

Anyway, the link to the Post story will direct you eventually to the answer. But don’t investigate until you’re sure of your effort. My guess is half of readers will solve it in under five minutes, and a quarter will take ten, and a quarter won’t get it and will peek. But the readers of this blog can hardly be considered the average “Internet.”

Now today’s lesson—and challenge. The order that you receive information matters in logic, hence in probability. Think of the Monty Hall problem. It was crucial for the solution that Albert said what he said when he said it. That’s the only hint I’ll give.

So our challenge is to make a superior probability puzzle out of this. I don’t want to post any of the ideas I have until later today (or tomorrow?) after you’ve had a chance to answer the puzzle. I’m betting we can come up with a good twist on this puzzle that makes it a true stumper—and not just something which fools reporters.

Update I see some wrong answers here and there. I’ll post a better-worded version of the solution here late this afternoon, after we’ve had more time to stew. Writing it took longer than thinking of it! (As usual.)


Both the problem and official solution are poorly worded, so I thought I’d clarify them here.

The setup would have been better written like this: “Cheryl wants to play cutesy and so tells Albert the month in which she was born but not the day, and she tells Bernard the day but not the month. She gives them 10 possible dates (month + day) and then lets the boys fight over her.”

If Cheryl told Bernard the day was 18 or 19, then because each of these days appears only once, Bernard would know the date. For instance, if she told Bernard the day was the 18th, Bernard would know the date was June 18.

Albert knows the month, and reasons that if the day were the 18th or 19th, Bernard would know the date, which means the month could only be May or June. But then Albert declares he knows Bernard doesn’t (or can’t) know the date. That must mean Albert knows the month is either July or August, since if it were May or June, it is then possible Bernard might know the date and Albert could not say with certainty Bernard is ignorant of the date.

Bernard hears Albert say that he, Albert, deduced Bernard could not possibility know the date based on the initial information provided by Cheryl. Bernard then reasons just as Albert did and deduces the month must be July or August.

Recall Bernard knows the day. If Bernard knew the day was the 14th, Bernard could only deduce, given Cheryl’s list, the month is July or August. So it can’t be the 14th because Bernard says he knows the date. If Bernard knew the day was the 15th or 17th, then he’d know the month was August, and if he knew the day was the 16th, he’d know the month was July. Whichever way, he deduces the date and tells Albert of his deduction.

And then, after hearing Bernard announce he knows the date, Albert, knowing the month, says he knows the date, too. The only way for this to be true is if the month were July, because if it were August, Albert would be left guessing the date is August 15th or August 17th.

Thus Cheryl’s birth date is July 16th. Don’t forget there are four perspectives here: Cheryl’s, Albert’s, Bernard’s, and yours. Each person has different information at different times, and the order matters.

Probability Challenge

We can turn this into probability by expanding the choices. If we do it cleverly, the probability Cheryl’s birth date is such-and-such a date would be different for the two boys. Seems to me Cheryl, the minx, would like it better that way.

I haven’t had the time to think of any solutions where both would have different probabilities, but it’s easy to imagine setups where they both have the same: i.e., simply by expanding the (different) days in July and August.

Space’s 10 Mistakes People Make When Arguing Science

Picture cropped from the article.

Picture cropped from the article.

Reader Ken pointed us to Space.com’s click-me-click-me! article “The 10 Mistakes People Make When Arguing Science”. Since everybody loves numbered lists, I’ll duplicate their efforts here, using their mistakes. Isn’t it curious most are probabilistical (yes, probabilistical)?

1. Wait! That’s Just One Study!

In some cases, one study is enough. In some, none is plenty. Gort the caveman is out in the rain and sees a bolt of yellow light blast a stand of trees. Tree catches fire and a deer which had been standing nearby keels over. Gort reasons, “Nasty business those bolts, what? what?” He then advises Qweeloc, another member of the tribe, to keep clear. Gort needn’t know how or why the bolt bangs to understand that its deadliness. And all Qweeloc has to do is trust Gort.

A lightning bolt is a physical thing, and physical things are are easy to understand. Consider: Gort brought the skin of the dead deer home to Fuh, his darling mate, thinking she would be well pleased. She wasn’t and complained about the smell. This perplexed Gort extremely. He reasoned, “Sometimes my gifts please her, and sometimes they don’t. I can’t figure out what works.” And that’s because human behavior is hideously difficult to predict.

Update Since Gort only has a sample of 1, he’d never get a wee p-value, thus he could never prove that lightning kills, and his paper would be rejected.

2. Significant Doesn’t Mean Important

But human behavior is shockingly to easy to mistakenly claim to have been understood. If Gort would have treated this latest gift as a part of a “random” experiment for which he calculated a p-value, he could have claimed “statistical significance”. He then would have been able to publish a peer-reviewed drum song in Caves and Hovels and secured his success as a researcher.

“My results are statistically significant!” sounds juicy and sounds like truth. It isn’t. Significance means finding a wee p-value and nothing else. This is why your better sort of statisticians says Die P-Value, Die Die Die.

3. And Effect Size Doesn’t Mean Useful

Knowing that having a second weekly doughnut “doubles the risk” of splenetic fever is frightening. Until you realize that the risk goes from 1 to 2 in 10 million. As I wrote:

Not a trick question: what’s the difference between a risk of one in ten million and one of two in ten million? The official answer is “Not much.” Though I would also have accepted “Almost none”, “Close enough to be the same,” and “Who would care?”

Effect sizes communicate risk in terms of parameters, little mathematical bits inside models which are of no interest to anybody who wants to know the risk of a real thing. The risk of real things is necessarily smaller than the risk in the reported effect sizes. Effect sizes are a sure way to produce over-confidence. Selling fear is a risky—but profitable business.

4. Are You Judging the Extremes by the Majority?

A scientist stuffs a rat living in an artificial environment full of chemical X and watches it develop cancer. The researcher then suggests humans “exposed to” X will, too. A reporter will then write a headline, “X Causes Cancer! When Will The Government Do Something?”

The government then does something. But the government also ignores hormesis, which describes the benefit low doses of X provide. Worried about your kid developing asthma? Some say, “Send him outside to play in the mud and dust and dirt.” Outside, incidentally, is that vast uncovered place far from “devices.”

5. Did You Maybe Even Want to Find that Effect?

Every working scientist knows about confirmation bias. Just as every working scientist knows it only happens to the other guy.

“Still—it is possible…for a historian or a scientist or, indeed, for any thinking man to present evidences, from a proper employment of sources, that are contrary to his prejudices, or to his politics, or indeed to inclinations of his mind. Whenever this happens, it manifest itself in his decision to present (which usually means: not to exclude) evidences not supporting his ideas or theses.” So says John Lukacs in his At the End of an Age. He calls the class of behavior of scientists like this, not objectivity, but honesty.

How much of it do we have in science today?

6. Were you Tricked by Sciencey Snake Oil?

The science is settled! It’s a Consensus! And what says Science better than Consensus?

Read: The Consensus Fallacy. Or, even better, read The Consensus In Philosophy.

7. Qualities aren’t Quantities and Quantities aren’t Qualities

How much do you feel, on a scale of -3.4 to 117 2/3, that bulleted lists make you happy? Be precise. I intend to collect the answers of this instrument and write a scientific paper on how I feel about your feelings.

Improper and absurd quantification is a plague on science. See this growing list of Asinine Uses of Statistics.

8. Models by Definition are Not Perfect Representations of Reality

Don’t tell physicists who are on the hunt for the multiverse this. It will make them sad. And that could be a hate crime.

See this video on the love of theory. And remember what that fairly often sober statistician said, “The love of theory is the root of all evil.” Also see Theory confirmation and disconfirmation.

9. Context Matters

Thinking that it doesn’t is one of the great causes of over-certainty. Scientific statements are usually (mostly? always?) probabilistic, and all probability is conditional, which is another way to say context matters. Read There Is No Such Thing As Unconditional Probability or the Monty Hall Problem

10. And Just Because It’s Peer-Reviewed Doesn’t Make It Right

Peer review is, at this point in our watered down world of science, nearly useless. If you don’t believe this, look at the asinine list or at these peer-reviwed studies.

On Remaining Quiet In Time Of War

Defend us in battle.

Defend us in battle.

Those who are unwilling to hold this is true will be subject to those who will. Those who cannot defend their beliefs must accept the rule of those who do. Those who remain quiet in the face of adversity must remain mute when their adversary triumphs.

A religion unwilling to say it is the only way to truth must wither. And anyone calling for dialogue wants you to change your belief.

Dogma has a bad reputation, but only the word itself, not the notion. We’re all after the truth.
Now some “feel” they have the truth, but others think it. And it’s those suffering from “feelings” who are most likely to have missed the mark. Feelings are too susceptible to whim and manipulation and to unregulated passions to be relied upon.

But what about two parties who each claim to think the truth? Both can’t be absolutely right, though both might be absolutely wrong, or both might grasp portions of the truth. Compromise in only the latter case is possible, and only in those areas which both parties acknowledge are uncertain. No agreement will be found in any matter which one party claims is dogma.

Truth exists and we can know it. One divided by one equals one no matter what. Yet I once knew a lady (regular readers will recall) who insisted one divided by one was zero “because when you divide one by itself, nothing is left.” Only one of us could be right. No compromise was possible. “Dialogue” would be absurd.

What to do about the lack of compromise in this or any argument is an entirely separate matter from what the truth is. The consequences of this lady’s mistake were trivial and there was no clear benefit for me (or for anybody) to marshal forces to defeat this incorrect belief. Her mistake was harmless and she was unlikely to gain supporters. And even if she was, it wouldn’t come to much. Math is of little consequence to most.

Yet some mistaken beliefs cause grief, both to their holders and to others. In those cases there is conflict, and in the keenest differences, there is war.

War does not have to be material; it can be, and most often is, spiritual. We have been, are, and will continue to be in a spiritual war. Such is the fate of mankind. If you disagree with this, you and I are in conflict, hence you confirm the claim. There is no escape. There are only sides and you must choose.

Our present war, currently spiritual but threatening to become material, is sex. Sin exists. Not crime: sin. Fornication is wrong, a sin. Some think it right; most others feel it is. I’d bet the word is now so foreign to most of you that it seems something dredged up from ancient history, a strange custom now happily abandoned, much like the wearing of corsets.

Divorce is wrong, a sin. One man, one woman make a marriage, and for life. A society which abandons that principle must eventually fail. How harsh that sounds! How cruel! How judgmental! But its harshness signals the truth that those who hold divorce is immoral are at war with those who think (and feel) it is not.

One man cannot marry another, nor can one woman wed a second. Nor can one man marry his dog, nor can a woman wed herself. Everybody forgets, despite constant and raucous reminders, that marriage is not only a spiritual union between a man and wife for life, but also between that couple and society. This is why the war has been so vehement.

If marriage were only an impermanent contract between any group of people (why insist on just two?), then few would care what anybody did. But because marriage is between a couple and society, those who claim to be married must insist that all others agree these unions are valid.

A man pretending to be a woman is not a woman, but is only a man pretending to be a woman. There is no such thing as a “sex change” operation, nor can anybody begin, let alone conclude, a “transition” to a different sex. Further, both of these circumlocutions are horrible abuses of the English language.

Our society is at war. One side, sensing victory, gleefully calls for the use of force against its enemy, while the leadership of the other is in full retreat, leaving the foot soldiers to fend for themselves. Despair would therefore be indicated, except that the foot soldiers have Truth on their side. And they won’t shut up.


I know thy works, that thou art neither cold nor hot: I would thou wert cold or hot. So then because thou art lukewarm, and neither cold nor hot, I will spue thee out of my mouth.

Summary Against Modern Thought: God’s Understanding Is His Essence

This may be proved in three ways. The first...

This may be proved in three ways. The first…

See the first post in this series for an explanation and guide of our tour of Summa Contra Gentiles. All posts are under the category SAMT.

Previous post.

We’re entering an arcane patch, a thicket of argument to prove a technical point about God’s intelligence. I’ll skip around some of this material, but not all of it. We need it to get to the punchline, which is coming in 15 chapters, that God is Truth. From Truth we move to Good and Evil. So this thicket is well worth punching through, though I admit that some attentions will flag.

Chapter 46 That God Understands by Nothing Else but His Essence (alternate link)

[1] FROM what has been proved above it is made evident that the divine intellect understands by no other intelligible species but the divine essence.

Notes Think of “intelligible species” as the how of how God thinks. What Aquinas is doing is showing, or rather illuminating, a point he already proved, that God is simple (in the meaning of that technical term) way back starting in Chapter 18.

[2] For the intelligible species is the formal principle of the intellectual operation; even as the form of every agent is the principle of that agent’s proper operation. Now the intellectual operation of God is His essence, as we have shown.[1] Wherefore something else would be the principle and cause of the divine essence, if the divine intellect understood by some intelligible species other than His essence: and this is in contradiction with what has been shown above.[2]

Notes And this is the main point. A car operates by being in the form of a car, yes? And God operates by His form, His essence. A weak analogy would be to say that if God had a brain by which His intelligible species operated, He would have parts, which we have already proved He cannot. And we’d also have to explain how this “Brain of God” moved God’s intelligence around separate from His will. Sort of like how materialists have to prove the “Brain of Man” moves your intellect about separate from your will. The next two arguments illuminate this.

[3] Again. The intellect is made actually intelligent by the intelligible species: just as sense is made actually sentient by the sensible species. Hence the intelligible species is compared to the intellect as act to potentiality. And consequently if the divine intellect were to understand by a species other than itself, it would be in potentiality with respect to something: and this is impossible, as we have proved above.[3]

[4] Moreover. An intelligible species that is accessory to the essence of the intellect in which it is, has an accidental being: for which reason our knowledge is reckoned among the accidents. Now in God there can be no accident, as proved above.[4] Therefore there is no species in His intellect besides the divine essence…

Notes In [3] there is the proof, if you like, that animals are sentient, which means having the capability of sensing. This is only worth pointing out because of the modern understanding that all sentient creatures, including humans, are alike. This is so, but humans are not just sentient, but are also rational, possessing an intellect, and animals do not.

The second point is the ever-necessary distinction between act and potential. Only something in act, or that has actuality, can move a potential to act, i.e. can cause something to happen. In God there is no potentiality, in us there is. The second translation of [4] (linked above) puts it in better modern English, “an intelligible species in the intellect that is other than the intellect’s essence has an accidental being, which is why our knowledge is numbered among the accidents.” Some (I don’t think we can say all) of the knowledge we have is not essential to our being, but all of God’s knowledge is His essence.

[6] Moreover. God’s act of intelligence is His essence, as we have proved.[5] Therefore if He understood by a species that is not His essence, it would be by something other than His essence. But this is impossible.[6] Therefore He does not understand by a species that is not His essence.

Notes And that, I think, is enough for today. Review the terms act and potential, or actuality and potentiality, and form and accident. (All the posts in this category are found in the link at the top.) Next week it gets harder.


[1] Ch. xlv.
[2] Ch. xiii.
[3] Ch. xvi.
[4] Ch. xxiii.
[5] Ch. xlv.
[6] Ch. xxii.

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