Category Archives: Basic Set Theory and Logic

Alice in Wonderland and probability stuff !!

Lewis Carroll of Alice’s Adventures in the Wonderland was a mathematician in Cambridge. He had posed the following problem in one of his books:

A box contains a handkerchief, which is rather white or black. You put a white handkerchief in this box and mix up the contents. Then you draw a handkerchief, which turns out to  be white. Now, if you draw the remaining handkerchief, what is the probability of this one being white?

How likely is that you would enjoy such questions ?! 🙂

Nalin Pithwa.

Gruffs

The reporter for the local Gazette was at the national dog show yesterday. Unfortunately, he also had to report on quite a few other events on the same day. He made some notes when he was at the dog show which are shown below, but the editor of the Gazette has asked for a list of the 26 finalists and the positions in which they finished overall. Using the reporter’s notes below, see if you can construct the list for the editor, as the reporter didn’t have time to make a note of the overall positions.

The Afghan Hound finished before the Alsatian, the Poodle and the Beagle. The Beagle finished before the Bull Mastiff and the Labrador. The Labrador finished before the Poodle and the Pug. The Pug finished before the Alsatian, the Dobermann Pinscher, the St. Bernard, the Sheepdog and the Griffon. The Griffon finished before the Sheepdog. The Sheepdog finished after the Poodle. The Poodle finished before the St. Bernard, the Collie, the Pug, the Griffon, the Alsatian and the Dobermann Pinscher. The Dobermann Pinscher finished after the Alsatian. The Alsatian finished after the Chow. The Chow finished before the Afghan Hound, the Bulldog, the Chihuahua, the Poodle, the Beagle and the Dachshund. The Dachshund finished before the Spaniel. The Spaniel finished before the Foxhound.

The Foxhound finished before the Labrador. The Labrador finished after the Whippet. The Whippet finished before the Great Dane. The Great Dane finished after the Bull Terrier and before the Chow. The Chow finished before the Bull Mastiff. The Bull Mastiff finished before the Foxhound. The Foxhound finished after the Yorkshire Terrier. The Yorkshire Terrier finished before the Greyhound. The Greyhound finished before the Dachshund. The Dachshund finished after the Kind Charles Spaniel. The King Charles Spaniel finished before the Bull Mastiff, the Greyhound, the Pug, the Chihuahua and the Afghan Hound. The Afghan Hound finished after the Dalmatian. The Dalmatian finished before the Pug, the Labrador, the Poodle and the Collie.

The Collie finished before the Pug. The Pug finished after the Retriever. The Retriever finished before the Bull Terrier, the Chow, the Yorkshire Terrier and the Whippet. The Whippet finished before the Chow, the Spaniel, the Dalmatian and the Yorkshire Terrier. The Yorkshire Terrier finished after the Bulldog. The Bulldog finished before the Chihuahua and the Dalmatian. The Dalmatian finished after the Great Dane. The Great Dane finished before the Yorkshire Terrier. The Yorkshire Terrier finished before the Collie, the King Charles Spaniel, the Spaniel and the Dalmatian. The Dalmatian finished after the Spaniel. The Griffon finished before the Alsatian. The Alsatian finished after the Sheepdog. The Griffon finished after the St. Bernard. The Greyhound finished before the Chihuahua. The Chihuahua finished before the Dachshund. The Bull Terrier finished before the Whippet and the Yorkshire Terrier. The Afghan Hound finished before the Pug and the Foxhound.

So, if u try this, it might ‘hound’ u !!! Good puzzles ain’t easy! But, such puzzles help u develop the habit and power of sustained thinking on a problem for long hours. 

Nalin Pithwa.

I have a sweet tooth !!!

  1. Tracey will not get the chocolate-covered mints unless Neil has the plain mints.
  2. Alan will not get the toffee unless Robert has the mint-flavoured toffee.
  3. Neil will not get the plain mints unless Alan has the mint-flavoured toffee.
  4. Robert will not get the toffee unless Tracey gets the plain mints.
  5. Robert will not get the chocolate-covered mints unless Tracey gets the toffee.
  6. Tracey will not get the toffee unless Neil gets the chocolate.
  7. Robert will not get the mint-flavoured toffee unless James gets the plain mints.
  8. Alan will not get the plain mints unless Tracey gets the toffee.
  9. Tracey will not get the plain mints unless Alan gets the chocolate-covered mints.
  10. Alan will not get the mint-flavoured toffee unless Tracey gets the chocolate-covered mints.
  11. James will not get the plain mints unless Tracey gets the toffee.
  12. Neil will not get the chocolate unless Alan gets the chocoate-covered mints.
  13. Roberts will not get the plain mints unless James gets the toffee.
  14. James will not get the toffee unless Tracey gets the plain mints.
  15. Neil will not get the toffee unless Tracey gets the toffee.
  16. Tracey will not get the plain mints unless Roberts gets the toffee.
  17. Alan will not get the chocolate-covered mints unless James gets the plain mints.

Who will get what ?

Oh, to be precise, I do not have a sweet tooth. I have sweet-teeth! Ich habe eine stucke schokolade sehr gehn! 

Auf wiedersehen.

Nalin Pithwa.

People’s Pets

Consider the following:

  1. Five men each have different first names and different surnames, have five different pets and live at five different addresses. All five pets have a different name.
  2. Tom’s surname is Williams and the fish is not  called Spike, Benson or Rodney.
  3. Harry has a pet cat and the budgie is called Percy.
  4. George’s surname is not Hudson or Smith.
  5. Mr. Thompson owns the dog and the owner of the rabbit lives in Pine Avenue.
  6. The cat is not called Benson and one of the five men has a pet called Fred.
  7. Mr. Anderson does not live in Cedar Road.
  8. Mr. Hudson lives in Willow Street.
  9. Mr. Anderson owns a pet called Percy and John lives in Cedar Road.
  10. Bill’s pet is called Rodney and is not  the dog; the owner of the fish lives in Maple Grove.

So, who lives in Chestnut Crescent and what is the name of their pet?

-Nalin Pithwa.

All The Twos

Alex, Brad, Colin, Doug, Eric and Frank had a race to school from the bus-stop and then another race from school to the bus-stop. In the first race, Brad wasn’t last and Eric finished before Colin, Frank wasn’t first but finished before Doug, Brad finished before Frank but after Alex and Colin. Alex finished four places ahead of Doug. In the second race, Alex finished two places ahead of Doug and before Eric who in turn was two places ahead of Doug and before Eric who in turn was two places behind Colin. Alex wasn’t first and nor was Brad who finished before Eric. Frank finished one place ahead of Doug.

From the information given:

  1. Which two boys finished in a better position in the first race than in the second race?
  2. Which two boys finished in the same position in the second race as they did in the first race?

Let me see which of my student(s) is first in this !

Nalin Pithwa.

Fun with English !!!

Done what ?

Arthur said Dave did it, Dave said Bill did it, Bill said Charlie did it, Charlie said that he didn’t do it and Eddie confessed that he did it. If Arthur didn’t do it, one of the five of them did do it and only one of them is telling the truth, who did do it?

🙂 🙂 🙂

Nalin Pithwa

Geometry problems for Pre-RMO

Practice Quiz:

  1. Prove that the median of a triangle which lies between two of its unequal sides forms a greater angle with the smaller of those sides.
  2. Point A is given inside a triangle. Draw a line segment with end-points on the perimeter of the triangle so that the point divides the segment in half.
  3. If the sides of a triangle are longer than 1000 inches, can its area be less than one inch?

You are most welcome to share your answers,

Nalin Pithwa

Richard’s (logical) paradox

In 1905, Jules Richard, a French logician, invented a very curious paradox. In the English language, some sentences define positive integers and others do not. For example, “The year of the Declaration of Independence” defines 1776, whereas “The historical significance of the Declaration of Independence” does not define a number. So what about this sentence: “The smallest number that cannot be defined by a sentence in the English language containing fewer than 20 words.” Observe that whatever this number may be, we have just defined it using a sentence in the English language containing only 19 words. Oops.

A plausible way out is to say that the proposed sentence does not actually define a specific number. However, it ought to. The English language contains a finite number of words, so the number of sentences with fewer than 20 words is itself finite. Of course, many of  these sentences make no  sense, and many of those do make sense don’t define a positive integer — but, that just means that we have fewer sentences to consider. Between them, they define a finite set of positive integers, and it is a standard theorem of mathematics that in such circumstances there is a unique smallest positive integer that is not in the set. So on the face of it, the sentence does not define a specific positive integer.

But logically, it can’t.

Possible ambiguities of definition such as “A number which when multiplied by zero gives zero” don’t let us off the logical hook. If a sentence is ambiguous, then we must rule it out, because an ambiguous sentence doesn’t define anything. Is the troublesome sentence ambiguous, then? Uniqueness is not the issue:  there can’t be two distinct smallest-numbers-not-definable- (etc.), because one must be smaller than the other.

One possible escape route involves how we decide which sentences do or do not define a positive integer. For instance, if we go through them in some kind of order, excluding bad ones in turn, then the sentences that survive depend on the order in which they are considered. Suppose that two consecutive sentences are:

  1. The number in the next valid sentence plus one.
  2. The number in the previous valid sentence plus two.

These sentences cannot both be valid — they would then contradict each other. But, once we have excluded one of them, the other one is valid, because it now refers to a different sentence altogether.

Forbidding this type of sentence puts us on a slippery slope, with more and more sentences being excluded for various reasons. All of which strongly suggests that the alleged sentence does not, in fact, define a specific number — even though it seems to

With thanks to Prof Ian Stewart for this gem from his Cabinet of Mathematical Curiosities,

Nalin Pithwa

G H Hardy on Set Theory

In these days of conflict between ancient and modern studies, there must surely be something to be said for a study, which did not start with Pythagoras and which will not end with Einstein, but is the oldest and the youngest — G H Hardy

Alien Encounter: test your logic skills

Alien Encounter.

The starship Indefensible was in orbit around the planet Noncompostments, and Captain Quirk and Mr. Crock had beamed down to the surface.

“According to the Good Galaxy Guide, there are two species of intelligent aliens on this planet,” said Quirk.

“Correct, Captain —- Veracitous and Gibberish. They all speak Galaxic, and they can be distinguished by how they answer questions. The Veracitors reply truthfully, and the Gibberish always lie.’

‘But physically —‘

‘— they are indistinguishable, Captain.’

Quirk heard a sound, and turned to find three aliens creeping up on them. They looked identical.

‘Welcome to Noncomposimentis,’ said one of the aliens.

‘I thank you. My name is Quirk. Now, you are…’ Quirk paused. ‘No point in asking their names,’ he muttered. ‘For all we know, they will be wrong.’

‘That is logical, Captain,’ said Crock.

‘Because we are poor speakers of Galaxic,’ Quirk improvised, ‘I hope you will not mind if I call you Alfy, Betty and Gemma. As he spoke, he pointed to each of them in turn. Then, he turned to Crock and whispered, ‘Not that we know what sex they are, either.’

‘They are all hermandrofemigynes,’ said Crock.

‘Whatever. Now, Alfy: to which species does Betty belong?’

‘Gibberish.’

‘Ah, Betty: do Alfy and Gemma belong to different species?’

‘No.’

‘Right…Talkative lot, aren’t they? Um…Gemma: to which species does Betty belong?’

‘Veracitor.’

Quirk nodded knowledgeably.’Right, that’s settled it, then!’

‘Settled what, Captain?’

‘Which species each belongs to.’

‘I see. And those species are —?’

‘Haven’t the foggiest idea, Crock. You’re the one who’s supposed to be logical!’

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So, dear students, have some fun!!

More later,

Nalin Pithwa