Logic Puzzles, Clue, and Justified True Belief
I. The Power of Demonstrative Logic in Action.1
There are two kinds of reasoning… demonstrative reasoning and plausible reasoning… they do not contradict each other; on the contrary, they complete each other. In strict [demonstrative] reasoning the principal thing is to distinguish a proof from a guess, a valid demonstration from an invalid attempt. In plausible reasoning the principal thing is to distinguish a guess from a guess, a more reasonable guess from a less reasonable guess.
George Polya, Mathematics and Plausible Reasoning, Vol. 1, Preface, p. vi
There’s an old joke that the hardest part of law school is getting admitted. This is in no small part due to the Law School Admissions Test – or LSAT. It’s a juiced up SAT and one of the more intriguing, and difficult, parts of it is the logic puzzle section(s). For those reading this who have done it, you’ll know exactly to what I’m referring. For those non-lawyers, let me share the pain and ‘splain.
Below is one of the 4 puzzles that appeared on the June 2007 LSAT.
Exactly three films—Greed, Harvest, and Limelight—are shown during a film club’s festival held on Thursday, Friday, and Saturday. Each film is shown at least once during the festival but never more than once on a given day. On each day at least one film is shown. Films are shown one at a time. The following conditions apply:
On Thursday Harvest is shown, and no film is shown after it on that day.
On Friday either Greed or Limelight, but not both, is shown, and no film is shown after it on that day.
On Saturday either Greed or Harvest, but not both, is shown, and no film is shown after it on that day.
What follows this short puzzle is a series of questions based upon the given information. For example, the first question:
Which one of the following could be a complete and accurate description of the order in which the films are shown at the festival?
(A) Thursday: Limelight, then Harvest; Friday: Limelight; Saturday: Harvest
(B) Thursday: Harvest; Friday: Greed, then Limelight; Saturday: Limelight, then Greed
(C) Thursday: Harvest; Friday: Limelight; Saturday: Limelight, then Greed
(D) Thursday: Greed, then Harvest, then Limelight; Friday: Limelight; Saturday: Greed
(E) Thursday: Greed, then Harvest; Friday: Limelight, then Harvest; Saturday: Harvest
The problem isn’t that the puzzles are impossibly hard – they can all be “figured out” – which is another way of saying that with the information provided, one can use demonstrative reasoning to answer all of the questions correctly. The real “problem” is that you’re on the clock, there are 30 questions (IIRC) from about 4-6 different scenarios, and you have to be able to move through the questions quickly, but without making an inferential mistake. Just to be clear for emphasis: plausible reasoning – reasoning by plausibility (ie. inductively), or by analogy, or some other way – plays no part in getting the correct answers on the LSAT.
LSAT problems are purely an exercise in demonstrative (deductive) reasoning. To repeat: in demonstrative reasoning, the conclusions follow from the premises with certainty; in plausible reasoning, the conclusions follow from the premises with probability.
Many people may be familiar with Sudoku puzzles, which are – at heart – another kind of puzzle that relies entirely upon demonstrative reasoning from a few premises – in Sudoku, it’s that each box, and each row and column, must all contain a set of digits 0-9, (i.e. with no repeats or omissions). All of Euclidean geometry – its beauty, elegance, and power – rests on a few definitions and axioms: point, line, and a plane, and perpendicular and parallel lines. Purely demonstrative reasoning. For example, look at the middle box in the puzzle: it’s almost all filled in except for 2, 6, and 8. Now glance at the box just to its left, and spot the number 8 in the top row. Given that there can only be a single “8” per row, column, and within a box, the “8” in the Middle Box has only one possible home: dead center. What I just did there is purely demonstrative reasoning.
The range of games and puzzles that human beings have designed with demonstrative reasoning at the heart of it (which includes all card games) probably deserves its own study, but CLUE is one of my childhood favorites. It is possible from well-chosen questions to other players and meticulous note-taking to cipher that Professor Plum did it in the Conservatory with the Wrench. (Not always). As anyone who has played CLUE knows, there are 3 categories of cards: Rooms, People/Suspects, and Weapons. The cards get shuffled, and three cards – one from each category – are picked “blind” and then placed into a cool little card-sized envelope. The envelope now contains WhoDunnit! (Cue the organ music) With what weapon and in which room. You roll a die and move a piece around the board – a layout of the house where the murder occurred, and propose (out loud) a suspect, room, and weapon to one of the other players, who MUST show you one (but only one) of the cards if they have any of them. Again: by (mostly)2 demonstrative reasoning, with no inferential errors, you can figure out what cards are in the envelope and win the game.
Speaking of murder mysteries, Sherlock Holmes is sometimes held up in popular culture as a model of “deductive reasoning” when, in point of fact, many of Holmes’ conclusions were the result of inferential chains of plausible reasoning, not demonstrative reasoning. Holmes would – well, perhaps more correct to say Conan Doyle through his narrator Watson – would deliver post hoc soliloquies that chained together various pieces as if they were steps of a proof in Euclidean geometry, but many of Holmes’ best feats are strings of more or less plausible chains of inference.3
II. Justified True Belief is Knowledge.
One of the critical points to understand about the above examples is that in all cases, as long as you don’t make an inferential mistake – i.e. an unjustified leap in logic – you should be able to get the correct answer eventually. To be clear, this is like one of those exceedingly complicated mazes or word search puzzles as a kid: the answer is there, you just have to grind your way through it.
So what does it mean to make an inferential mistake or an unjustified “leap in logic”? In demonstrative reasoning (i.e. formal logic), like the LSAT puzzle above, you are given a piece of evidence – a proposition, a clue, or call it A – and from that certain other things are implied or can be logically derived within the overall set of rules under which you’re operating. As long as you adhere strictly to only that which is implied by each of the propositions – you should be able to figure out the order of all of the plays on each day of the festival (as per the LSAT example). Or whether it was Colonel Mustard or Miss Scarlet whodunnit in CLUE. An unjustified inference is one that does not necessarily follow from the proposition from which it is derived. If you were a homicide detective and showed up to examine the dead body in a suspect’s living room, there might be a number of conclusions you could reach given the body’s position, clothing, bruising, etc., but “what is my cat’s favorite toy” isn’t a conclusion that could be rationally extracted from that scenario.
This is where grids, charts, and tables, can sometimes help illustrate the limits of what is implied by a particular proposition and and what is not. Symbolic logic – turning statements into something more akin to mathematical equations – and then using algebraic manipulation and other mathematical techniques to (potentially) reveal relationships between the propositions that might not otherwise be obvious is a tool that would be taught in a world that properly educated its people about demonstrative and plausible reasoning.
This brings is into an interesting intersection of philosophy, mathematics, science and law: when is a belief justified and when is it not? Prof. William Briggs cheekily begins his 2022 book, “Everything you Believe is Wrong,” (italics in original) with this, on p.1:
Let’s be careful with the title. It’s not what you know that is wrong, but what you believe.
This is because you can only know what is true, you can never know what is false – but you can believe anything.
Is this true? Or just a bit of grammar pedantry? Prof. Briggs is expressing his view of a much deeper philosophical argument4 about how we “know” what we know – epistemology – and how we can claim to have acquired “knowledge” over mere “belief”. “Knowledge” is a “success word” in that it implies something more substantive than just having a recordation of a particular fact in your head.
For example, how many times have you heard the lyrics to an old song you loved years later only to find out that you had been singing complete gibberish ad alta voce – in front of God and everyone – for decades? If someone had asked you at the time if you “knew” the lyrics to the song, what would you have said (you little liar)? You didn’t know them; you thought you knew them. You only believed you knew them. (This isn’t a proof, just something for you to think about).
Belief is a subjective state of mind – a person’s attitude toward a proposition – while knowledge implies an objectively verifiable state of the evidence. Linguistically, we shall operate on that premise (for now) and allow philosophical skeptics to suspend their (dis)belief on my promise that we can argue about it in the comments and may delve into it at some point down the road. But as a general matter, I should like to point out that “other” people believe some crazy things with 100% certainty while being completely wrong. Knowing something does also have its own subjective component, too, in the formula that “Justified true belief is Knowledge.”
It means that in addition to my subjective belief of its truth, the proposition under consideration is also objectively verifiable, and one additional (non-obvious) requirement: that my “knowing” is a product of some understanding of the relationship contained in the proposition, not just a fortunate coincidence or superstition. That last part is a mouthful, and it is my argument against Hume’s inductive skepticism – which we haven’t quite gotten to, but we’re brushing up against – so (for the moment) I’ll say the following about about “Justified True Belief”: hockey goalies who won’t wash their gear or minor league pitchers who wear women’s underwear underneath their uniform while on a streak don’t know what’s causing the streak in the way necessary to meet our definition of Knowledge as Justified True Belief. They have a belief, and it is true, but it is not “justified”.
Or, to come back to my Clue analogy: You are justified in your true belief that Professor Plum did it in the Library with the Candlestick if (a) that’s what cards are in the little envelope, (b) you believed those cards were in the little envelope, and (c) you did it by a process of demonstrative reasoning within the context of the game, i.e. asking questions of the other players and figuring out the correct answer.5 5
III. Justified True Belief in Law.
In many, many places in the law whether or not one is “justified” in doing or believing something can be the difference between criminal or civil culpability, admission of evidence or not, a motion’s success or denial. To be clear, this is not a case of equivocation – of me substituting a word like “justified” from one context (philosophy, mathematics) into a different one (the Law) and trying to steal a base. I am also not using the word in any moral sense at all (i.e. justification for a particular act in criminal or civil law).
I am asserting that the same inferential standard in philosophy regarding what is justified true belief is used in the law by jurists to make decisions about, among other things, what evidence a jury can or can’t see, or whether or not a person’s actions were “reasonable” under a particular set of circumstances or evidentiary propositions.
If you were to go on Westlaw and search for the phrase “a reasonable factfinder could conclude” you will find more than 10,000 cases return where a judge has used used that exact phrase in various trial and appellate court opinions. The first thing you might ask yourself is how an appellate court judge can “know” – have any “justified true belief” – about what a juror at some trial court in a different part of the country, a year or more earlier, could have concluded? What exactly is going on intellectually with that rhetorical device – the “reasonable factfinder”? The answer comes from the use of demonstrative reasoning, as in the LSAT, Sudoku, and CLUE examples above.
A “reasonable factfinder” – from the view of a properly trained appellate judge – can only draw conclusions that are justified by the evidence presented to them in court. A factfinder’s “reasonableness” is a direct result of what inferences he or she could possibly have drawn from the particular evidence under examination. This comes up all of the time in trials under Rule 104 of the Federal Rules of Evidence.
(a) In General. The court must decide any preliminary question about whether a witness is qualified, a privilege exists, or evidence is admissible. In so deciding, the court is not bound by evidence rules, except those on privilege.
(b) Relevance That Depends on a Fact. When the relevance of evidence depends on whether a fact exists, proof must be introduced sufficient to support a finding that the fact does exist. The court may admit the proposed evidence on the condition that the proof be introduced later.
FRE 104
FRE 104(a)’s “whether a witness is qualified” comes up in conjunction with the rule on expert testimony, Rule 702, and that is where judges can make or break a case with their decision of exactly who qualifies as an expert and what qualifies as expert testimony. FRE 104(b)’s rule on conditional admissibility deals directly with what evidence is “sufficient” to “support a finding” (by a “reasonable factfinder”) of a fact’s existence, particularly when the opponent of the evidence argues that the requisite conditional fact hasn’t been proven sufficiently. Both pieces of Rule 104 have been subject to clarification as to what standard must be met on such preliminary questions. See, e.g., Bourjaily v. United States, 483 U.S. 171 (1987) (holding that preponderance of the evidence is the standard for a judge to decide preliminary questions under Rule 104a).
- Professor Polya uses the term demonstrative reasoning, so I use that convention throughout, but deductive reasoning is the more common American English term. I also use his convention because the term “deductive” reasoning or logic implies its opposite, “inductive reasoning” and Professor Polya shows in his books that inductive inference is a subset – albeit a very large one – of the broader genus of plausible reasoning. ↩︎
- Some plausible reasoning can also help here, too – if you know your sister’s ‘tells’ when she lies, for example. ↩︎
- When I was a wee lad in the ‘70s, the Scholastic Books sales at school used to have a kid-version series named “Encyclopedia Brown” – where the reason that Encyclopedia Brown was able to solve the mystery – was explained in the “solutions” in the appendix at the end of the book. I kid you not. ↩︎
- See, for example, Graham Dawson in The Philosophical Quarterly, Vol. 31, No. 125 (Oct., 1981), pp. 315-329, titled “Justified True Belief Is Knowledge.” Available at: https://www.jstor.org/stable/2219402 ↩︎
- As opposed to a completely uninformed lucky guess – i.e. you always guess that set of cards every time you play Clue! and it happened to deal up this time! – a series of questions to a Magic 8-Ball, or which cards your cat previously sniffed. ↩︎
My engineer brain says: replace the LSAT with a sudoku.
In fact, replace all standardized tests with a sudoku. Two guys finish at the same time? Tie breaker with a Rubik’s Cube.
Nice shout out to Polya. “How to Solve It” is his greatest work.
Italics omitted intentionally to annoy Tonio and The Hyperbole.
https://en.wikipedia.org/wiki/How_to_Solve_It
I have a copy of “How to Solve It” alongside “Mathematics and Plausible Reasoning” (Vols 1 and 2).
Polya’s amazing for contributions in both mathematics and teaching.
LSAT clue #1 eliminates all choices but B and C. Clue 2 leaves only B as the correct answer.
Now here’s a question that’ll bake your noodle:
x+y = 10
x*y = 20
find x and y
two unknowns, two independent equations – it can be done
Also, a lot of lawyers aren’t that smart. Let me know when there’s a Nobel Prize for law.
A is false because Greed is not shown at all.
B is false because Greed and Limelight are shown together on Friday.
C is true
D is false because Harvest is not last on Thursday
E is false because Limelight is not last on Friday.
Damnit! I meant C!
It was a typo! A typo I swear!
You’re gonna make me do a quadratic equation?
The answer’s a little less than 3 and a little more than 7, for x & y.
Do I need to show my work?
5 +/- sq rt of 20 /2.
x = (5 + 2.236) = 7.236
y = (5 – 2.236) = 2.764
Lawyers (and software engineers) care about being exact except when they don’t.
https://www.youtube.com/watch?v=qHMlTHeW2P4
+1 loglog++ algorithm
https://www.youtube.com/watch?v=lJYufx0bfpw
I wish there were more attorneys who understood math beyond 1/3rd plus expenses.
Thanks for the series!
Of course he didn’t mean to.
It was supposed to be differential equations.
OK, here’s a big tech interview question, and you have 10 seconds to solve it:
circle plus triangle = 10
circle plus square = 20
triangle plus square = 14
what does circle + square + triangle equal?
NO GOOGLING!
https://www.youtube.com/watch?v=fP0ULf2k6xE
what does circle + square + triangle equal?
I would argue this is an ill posed question and the answer is undefined. What does the + operator mean in the context of ‘circle’, ‘square’ and ‘triangle’ objects? What properties of the objects does + operate on?
It takes more than ten seconds to
READit. Anyone who uses such questions is an idiot.*READ
Wrong tag
Took me more than ten seconds, but I think I got it.
Hint (rather than full potential spoiler): square = triangle + 10
C+T=10
C+S=20
T+S=14
C=10-T
C=20-S
10-T=20-S
-T=10-S
S=10+T
2T+10=14
2T=4
T=2
C+2=10
C=8
8+S=20
S=12
So 22
C + T = 10
C + S = 20
T + S = 14
Add all 3 together to get”
2C + 2S + 2T = 44
divide everything by 2 to get
C + S + T = 22
The trick: there is no need to solve for the individual values of C, S, and T. We only want to know their sum.
Students ought to be taught early on that equations can be added together just like regular numbers.
Good programming is about minimizing the necessary lines of code to accomplish the task.
Incorrect.
You need to know what is going on, not just the answer.
Of course you can treat the names as variables and treat it as a system of equations. But there are also operations on geometric objects that can be defined. Perhaps the ‘lesson’ of the problem – use clear variable names that are not prone to mis-direction or don’t make assumptions about the statement of a problem that are not explicitly spelled out. But it strikes me as more of one those “oooo, we’re so clever aren’t we” problem statements.
“Why are manhole covers round?”
“Because that’s the standard in this country. They’re not all round.”
The point of the question, and all other serious computer science questions, is to get the answer with the least number of steps.
There is no way the Allies could have tested every intercepted Enigma message against every possible machine setting. They had to take shortcuts.
https://www.nsa.gov/History/Cryptologic-History/Historical-Figures/Historical-Figures-View/Article/1621747/arthur-j-levenson/
***
In a 1999 PBS documentary about the decoding project, Mr. Levenson said the team at Bletchley sometimes deciphered the German messages before German forces in the field could read them. “If it was something hot,” he said, “it’d get out in the field before the German commander got his.” In one case, Mr. Levenson said, the team decoded a message from German military leader Erwin Rommel and determined that German tanks were converging at a spot in Normandy where U.S. paratroopers were planning to jump. “They were going to drop one of the airborne divisions right on top of a German tank division,” Mr. Levenson said in the documentary. “They would have been massacred.” At the last moment, plans were changed, and the paratroopers averted disaster.
***
shortcuts like smuggled code books and brute forcing iterations until something clicked to identify the configuration. The whole point of the bombes was to make the brute forcing human-scale possible.
“Least steps” is not the most efficient method in all cases. It all depends on where your bottleneck is. If you need to perform your handful of steps on huge data sets you’ll be chugging for days.
The point of these questions is smugness.
Part of the problem is recognizing that it is a system of equations stated in an unfamiliar way.
Almost anyone can learn how to solve obvious equations with well-known algorithms. That’s just following instructions.
Talent hits a target others miss. Genius hits a target others can’t even see.
https://www.youtube.com/watch?v=g_0RWFQaeAQ
So how you you gel that with the attitude of “Throw more hardware at it” that permiates tech?
What are C, T, and S in your system of equations? What is the ‘+’ symbol? The original problem didn’t have that notation. You’ve made assumptions about the problem statement without justification. How do you know that the circle, triangle, and square objects admit an additive operation? Those names have definite mathematical meanings that don’t admit a simple ‘+’ operator that is defined on the real numbers as you’ve assumed. If the problem statement included “assume circle, triangle, and square are real numbers:” then your system of equations is justified. They are NOT justified when you use variable names with well defined mathematical properties; that’s just pretend cleverness.
***
The point of these questions is smugness.
***
Wrong. Lateral thinking is real.
***
Carl Friedrich Gauss (1777 – 1855) is one of the world’s most famous mathematicians.
This story has been flying around for years… who knows if it’s really true or not!?
Gauss was about 9 years old — already a super genius (much like Wile E. Coyote.) His teacher hated math and hated Gauss (because he was so smart).
As usual, the teacher walked into the class and gave them a horribly tedious arithmetic problem. They were to work on it and not bother him.
Here was the day’s problem:
Add the integers from 1 to 100.
They got out their slate boards and chalk and started hammering away!
…
The teacher quickly noticed that Gauss was not writing — HA! He had him now!
“CARL! Why aren’t you working?!”
“Because I already know the answer.”
“Oh? You’re so smart — why don’t you share your answer with the class?”
“It’s 5050.”
“#*@#&*!”
It was true. Gauss had figured it out… In his head… At 9 years old… Do you hate him too?
Want to know how he did it? It’s a trick!
***
Can you figure out the trick? I did when I was 9 too.
https://www.youtube.com/watch?v=ZmqRsa2Fkak
Simple you have 50 pairs that add up to 101. It’s not a lateral problem, it’s basic arithmatic.
Yes, but you’d be surprised how many “bright” math students and “highly educated” adults can’t see that immediately.
Math is at a loss because Galois refused to explain his equations and later died young in a duel. To him, many deep mathematical truths were obvious.
Add the integers from 1 to 100 is a well defined problem with your assumptions pretty explicit. Recognizing patterns and fast ways to get the answer is perfectly justifiable there.
My point is that there is no ‘answer’ to the original problem WITHOUT making the ASSUMPTION that words with well defined mathematical meanings have completely different meanings here; very humpty-dumpty. You haven’t solved the problem in the shortest way possible. You’ve solved a problem, subject to your assumed definitions of the parameters of the problem.
quibble = 49 pairs that add up to 100, 100 at the end, and 50 in the middle
or by arithmetic series, just multiply the number of terms by the average of the first and last term
Silly putzes!
I solved it long ago! I sit with my popcorn and watch you all fight it out! The answer cometh sooneth!
(I’ll continue munching (and thinking). Cubs are tied at 1 w the Padres in the top of the 9th.) Y’all are truly fantastic.
Really?
1+100
2+99
…
50+51
50 pairs that add up to 101. 101*50=5050
The point of the interview question is that it is not enough to solve problems; you must solve them quickly.
To put it another way, engineering is the art of doing well with $1 that which any idiot can do with $2.
quibble^2 – it’s perfectly valid to say 50 pairs that add up to 101: (1+100), (2+99),…(50+51)
In fact, it takes fewer steps since it’s just 50*101 vs 49*100+50+100. And we’ve already established that getting to the ‘answer’ with the fewest steps is the gold standard here.
Robustness in making sure your solution is solid and won’t fail is a more important element of engineering. If it’s worth engineering it’s worth over-engineering.
Fair point. Many paths to the top of the mountain.
For me, it was more intuitive to find pairs that add up to 100.
Overengineering* is always a mistake. That’s why the Abrams and Leopards sent to Ukraine are just pissing in the wind.
Even with trained crews, those tanks require too many spare parts and too much maintenance to be useful.
*I define overengineering as being more complicated than the bare minimum needed to accomplish the task.
https://www.youtube.com/watch?v=WJD2ObT3p44
The point of the interview question is that it is not enough to solve problems; you must solve them quickly.
That’s my point – you (the royal you) haven’t solved the problem, quickly or otherwise. You’ve solved a problem you defined by making assumptions about the nature of the problem statement. Maybe that was the right assumption. Or maybe it wasn’t and your solution, when implemented, is going to make a very big boom.
Whoever generated the problem statement valued faux cleverness and smugness above a clear statement about the issue/problem they wanted solved. Hopefully you’ve interpreted their faux cleverness correctly and implemented the correct solution, rather than dumped an encrypted database in clear text or generated an infinite loop of stabilizer trim.
Manhole covers are round for safety. You can’t misalign one and drop it into the hole.
dbleagle – they’re not all round.
To avoid controversies of language, there is an intelligence test called Raven’s Progressive Matrices. You don’t even need to know English to take it. It’s all about pattern recognition. I took it as part of my NSA code breaker application.
https://en.wikipedia.org/wiki/Raven%27s_Progressive_Matrices
https://platedlizard.blogspot.com/2023/07/nsa-zersetzung-gaslighting.html
That would be me, and I’m not going to fash myself for being a fish that can’t climb a tree and must therefore be stupid.
I’m going to have to put my thinking cap on, aren’t I?
*looks around for colander*
LSAT is c.
But the real question is… was it Colonel Mustard in the Conservatory with the lead pipe?
Ozy: love the links to the footnotes. If I missed that last week… then I’m a potato.
They don’t format the same when I copy them over, so occasionally I may gork it up, but I try to remember to get the footnotes formatted correctly. I’ve got #4 in the queue and I’ll add #5 tomorrow (I just finished that one).
Gotta get going on #6.
Thanks for the new series Ozy.
This is fantastic. I have to keep reading to comment further. First thoughts: Our brains work very differently. I can follow with no problem, but the puzzle-solving aspect is not a natural part of my repertoire.
I’d never thought of CLUE that way, and I haven’t played in far-too long to re-mastermind my sleuthing. It’s in my Top 5 Films list.
Admission: *I* did it, in The Hall, with M̵a̵d̵e̵l̵i̵n̵e̵ ̵K̵a̵h̵n̵ Mrs. White. (And in every other room. We’re murderous cunts.)
with M̵a̵d̵e̵l̵i̵n̵e̵ ̵K̵a̵h̵n̵
You wish.
Id prefer the maid
Don’t you dare touch Yvette.
We plan on murdering Her, in The Bedroom, with Pleasure. (She’s totally into it.)
Upon further review, yes indeed.
That movie looks to have been at the peak of her powers
hmm…
Yvette or Medusa?
https://www.youtube.com/watch?v=4DQgKom3Ox8
There’s a strain of weed so strong you can get high just from seeing it. It’s called “Medusa” because one look and you’re stoned.
Love is love.
Fucking like bunnies is like fucking like bunnies.
I mean maybe go through the the “abc’s” to make it last a little while…
/not an expert at being a cunning linguist
/ the “abc’s” have served me well.
Watson was the real detective and subject of Sherlock Holmes series.
Clue is a great game to introduced yoir kids to young and play often.
A Knives Out movie:
Character: Why isn’t it like Clue?
Detective (Daniel Craig): That’s a terrible game.
There were Clue books with other puzzles:
https://en.wikipedia.org/wiki/Clue_(book_series)
Somewhere in math purgatory, Martin Gardner is creaming his jeans.
6 suspects + 6 weapons + 9 rooms = 21 cards
3 cards are in the file, leaving 18 for the players
Assume 3 players to make the math easier.
Average hand includes a suspect, weapon, and room
Average player has 6 cards and thus can eliminate 2 weapons, 2 suspects, and 2 rooms immediately
There are more rooms than weapons or suspects, so the best strategy is to enquire about suspects or weapons.
https://www.youtube.com/watch?v=8nPqErDpwlo
Who plays with so few players?
You have to enquire about all three at once.
And since there are more rooms than suspects or weapons, a player will on average, have more room cards than either of the other two kinds.
I had that edition! Squeee!
Wherein I sigh with weary resignation:
Ones For Texas: 5th Grade Teacher Thomas Mayfield’s ‘Mathmatical Music’ Equals Success
https://www.youtube.com/watch?v=0JkNznoU2fg
meanwhile…
American mathletes come in 4th place in International Mathematical Olympiad
https://www.youtube.com/watch?v=FHyb7ctxe6c
And they hadn’t won in 20 years until the child of Singaporeans started coaching
[head desk]
https://www.youtube.com/watch?v=FHyb7ctxe6c
Whatever. Sing songs, make finger-paintings, do interpretive dance…anything except actual learning which is often neither easy nor fun
I put in the same link by mistake and my comment went into limbo.
compare and contrast:
Ones For Texas: 5th Grade Teacher Thomas Mayfield’s ‘Mathmatical Music’ Equals Success
https://www.youtube.com/watch?v=0JkNznoU2fg
American mathletes come in 4th place in International Mathematical Olympiad
https://www.youtube.com/watch?v=FHyb7ctxe6c
Learning can be easy and fun, but often it isn’t. There are consequences to ignoring that truth.
Happy national banana day!
https://www.nationaldaycalendar.com/national-day/national-banana-day-april-15
💁♂️🍌‼️
Put one in a tailpipe, or eat one with chopsticks. Enjoy.
https://www.youtube.com/watch?v=LdHKIx6smag
🎶🎶
Link text is wrong…should be 16…. 🤦♂️
Everyone is tarded these days.
Good morning, Sean, CPRM, Stinky, Ted’S., U, and Derpy (if you’re still awake,) and good afternoon, Pie!
That’s not what’s in my pocket …
I don’t like logic puzzles like that LSAT test, not because I can’t solve them, but because they claim to have only one true answer. Thinking ‘outside the box’ is a thing for a reason. Perhaps all films are shown all 3 nights, just not at the film festival. That is the kind of thinking I want from a lawyer I’d pay.
Kirk was the best Star Fleet Captain because he beat the Kobayashi Maru by not accepting the existence of a ‘no win scenario’.
He’s a lousy cheater and his complacency got Spock killed, just not killed enough to prevent several movies of declining quality (except for the one with the whales, that one wasn’t bad).
You want judges who will come up with creative ways to bend the law, like nationwide injunctions from district judges. :-p
https://www.youtube.com/watch?v=2hjZscXqwyg
There is just something so satisfying about Zegler/Disney/Snow White reboot getting absolutely destroyed.
She’s just young and narcissistic and stupid. She’ll probably grow to be old and narcissistic and stupid but maybe she’ll learn.
she just needs to go to space or something.
I don’t know if linking Letitia’s problems would be a case of colorectal narcotics, but I did say I wanted prosecutions of the corrupt and the fraudsters.
So that’s some good news.
The arrogance of prosecutors like her and Fanny what’s her last name is a sight to behold…or maybe they’re just stupid. Under a microscope while committing crimes in a political environment is a recipe for disaster.
So you’re saying Fanny is cunty?
The specific charge is the piece de resistance. 😄 (No, I don’t know how to put accents on my e’s in WordPress, nor do I particularly care to learn. Thanks anyway.)
It’s a keyboard issue, not a WordPress issue. On my smartphone or tablet, I can long press the E and get a choice of diacritical marks: è, ë, and so on.
🤔
The local rag doesn’t even mention Lititia’s legal troubles. Big events with state officials would be right in the wheelhouse for the capital district local news organization… if they were actual journalists.
NO END IN SIGHT Huge blow for locals in UK’s ‘third-world city’ where Army called in to tackle cat-sized rats & 21k TONS of rubbish
A major incident has already been declared by Birmingham City Council
https://www.thesun.co.uk/news/34450668/army-bin-crisis-rat-street-rubbish-birmingham-rayner/
English cities look like garbage dumps when everything is “working” I don’t want to know how it is when they don’t
From the internet
‘They’ve got rats big as cats
They’ve got rivers of mould
The smell goes right through ya
You’d best have a cold
When you first took the bins out
On a cold winter’s eve
You promised me dustcarts
were “waiting, you’ll see”….’
Here’s some background on Birmingham’s trash problem.
Activists sued the city because garbage collectors as a class were paid more than people in cleaner, safer, more female-dominated jobs. The sham judicial system ruled that the city was being discriminatory and awarded a grotesque amount of money to the plaintiffs and if the city didn’t equalize the pay rates between the garbage collectors and the clean, safe, indoor jobs, they were at risk of being sued again for grotesque amounts of money. Since they had no money, they cut the pay for the garbage collectors something close to the equivalent of $10k/yr, causing them to go on strike. The trash has since been piling up for five weeks.
In a sane system, the city and the judges should have told the plaintiffs “If you want to make as much as a garbage collector, you are free to be a garbage collector and do that miserable job.”
I know the background.
Some gross jobs require a little more pay to entice people to take them, it is true. Plumbers and the like aren’t doing it because they enjoy the smell.
@Pie – my fellow Americans may not, as I had only found out what had gone down recently, so I figured I’d share.
it happened abut the same time when english judges ruled discrimination because people mostly women who worked inside air condition stores in a mall were discriminated as they made less then the people mostly men who worked moving pallets in the giant distribution warehouses…
Those retail workers are free to apply for warehouse jobs and move pallets of cargo around.
I assume they are forklift certified and can regularly lift over 100 lbs?
An employment tribunal said store staff, who are predominantly women, should not have been paid at lower rates than employees in warehouses, where just over half the staff are male.
Lawyers for the shop staff described the judgement as “hugely significant” and the amount of back-pay owed could amount to more than £30m.
However, Next said it would appeal against the ruling.
Next argued that pay rates for warehouse workers were higher than for retail workers in the wider labour market, justifying the different rates at the company.
But the employment tribunal rejected that argument as a justification for the pay difference.
According to the tribunal’s ruling, between 2012 and 2023, 77.5% of Next’s retail consultants were female, while 52.75% of warehouse operators were male.
The tribunal accepted that the difference in pay rates between the jobs was not down to “direct discrimination”, including the “conscious or subconscious influence of gender” on pay decisions, but was caused by efforts to “reduce cost and enhance profit”.
It ruled that the “business need was not sufficiently great as to overcome the discriminatory effect of lower basic pay”.
The ruling means women such as Helen Scarsbrook, who has worked for Next for more than 20 years, are in line to receive thousands of pounds of compensation for the pay they missed out on.
If Helen wanted better pay than retail drone, she could have gone job hunting. Millions of people get better jobs while working retail every year.
Because, of course, wages are never based upon what you actually DO. 🙄
@GT – Dontcha know they repealed the law of supply and demand as it applies to labor?
The obvious move is for trash collectors to sue saying they should be paid like judges.
https://www.cincinnati.com/story/news/2025/04/15/first-wawa-open-in-ohio/82703313007/
Congrats Ohio.
Thanks! 😁 That one’s not far from me at all, but I’m pretty sure we’re set to get one in or close to Dayton. Worth a road trip?
I’m going to say not worth a road trip specifically to visit the Wawa.
Heck, I haven’t even checked out any of the new Sheetz locations in town, and apparently we’re also due to get a Buc-ee’s. We may soon be overly-convenienced around here.
How inconveniently overly convenienced.
Wah-wah (or wa-wa) is an imitative word (or onomatopoeia) for the sound of altering the resonance of musical notes to extend expressiveness, sounding much like a human voice saying the syllable wah.
I have nothing else to contribute
UK Supreme Court says legal definition of ‘woman’ excludes trans women, in landmark ruling
It’s time to burn things.
suh’ fam
whats goody
TALL POST TAX DAY CANS!
😑
Don’t remind me of tax day.
Start preparing now. It’s only 364 days until tax day.