Dec. 2

Recall that {x} is the fractional part of a number, that is how far it is from the larger integer less or equal to x.

Exercises:

• Try to compare in your head 192/133 and 79/53.
• If {x}=1/2, and {y}=1/2, what are possible options for {xy}?
• What are possible options for {x}+{-x}?

november 4

Homework:

• Is it possible to mark some cells in a 6x8 array with x's so that each row has exactly 3 x's and each column exactly 2 x's?
• Find all x such that |x|+|x-2|≤ 4.
• The side disks of a spool are twice the radius of the inner cylinder. If one pulls the line, which way the spool will roll? (There is no sliding.)

October 28

Homework

1. Buses stopping at Pete's home can take him either to math circle or to pool. Bus in each of these directions comes every ten minutes.
   Pete leaves home at random time (no one has a watch there!) and takes the first bus that appears. Yet, he goes to pool 4 times more frequently that to math circle. How is this possible?
2. Plot y=|x|-|x-2|.
3. Rope running along the equator (its length is 2πR, where R is the radius of the Earth - approximately 25,000 miles) is extended by 2 feet and lifted uniformly above the surface. Will a cat be able to squeeze beneath the rope?

October 7

Recall that |x| is the distance from x to 0, that is x if x is a positive number, and -x if it is negative.

Homework

• Rope was folded in 2 three times and cut through (resulting in 9 pieces). One of the pieces is 2 in, another 3in. What was the length of the rope? 
• Mark on the real line all the points a where 
1. |a-5|>3;
2. a+|a|=0;
3. |a-2|+|a-2|=3.

September 30

Homework:

• Mark on the number line all numbers x for which x/2+3 is an integer.
• Find the point that splits the interval [-6,8] in relation 2:5.
•  How many numbers from 000 to 999 have no even digits?
• Pete earns less than twice what John earns, and John earns less than half of what Dan earns. Who earns more, Dan or Pete?

September 23

Homework:

• Each next line is obtained from the previous one using a simple rule. Find it! Add 2-3 more lines using that rule:
  1
  1 1
  2 1
  1 2 1 1
  1 1 1 2 2 1
  3 1 2 2 1 1
  .....
  
• Out of 1000 numbers, starting with 0 and ending with 999, how many do not have digit 3 in them?
  
• Find midpoints of the intervals:
• [-7, 1000]
• [-1000, -7]

September 16, 2018

• Could chess knight jump over all squares of the $5\times 5$ board, visiting each just once?
  Answer - Yes!
• Could it do so and return to its starting point?
  Answer - No!
• A standard die is rolled to the right, towards you, to the left, away from you, returning to the original site; each time around an edge, without sliding. What will be the number on the top?
  How many times one will need to repeat this operation to return to the original orientation?
  
• How many different routes are there from A to F (moving always eastwards)?
  
  
  

Problem posted during our class on April 8:

There are four types of people:
boy knight and girl knave always tell truth;  boy knave and girl knight always lie.

There is one guilty person, which can be A, B, or C.  We want to find out who s/he is.

The judge has the following conversation with these three people:

Judge to A: Who is the guilty person?
A: He is a boy.
Judge to A: Is he a knight or knave?
A: He is a knave.

Judge to B: What do you know about A?
B: She is a girl.
Judge to B: Is she a knight or knave?
B: She is a knave.

Judge to C: Are you the guilty person?
C: (answers Judge privately, so we cannot hear if it is yes or no).
Judge: Now I know who is the guilty person.

Based on the above information we should figure out who is the guilty person.

Hints:

1. Which type of person (knight or knave, boy or girl) can A be?

2. Which type is the guilty person?

3.  If C answers yes, can we narrow down to one possibility regarding who is the guilty person?

4. If C answers no, can we narrow down to one possibility regarding who is the guilty person?

Logic problem discussed on 03/04/2018

There are four types: boy knight, boy knave, girl knight, and girl knave. 

Rule: Boy knight and girl knave always tell truth. Boy knave and girl knight always lie.

Setting: communication on the Internet, no face seen and no voice heard. Information only comes from the text message. 

Problem: determine the type of person based on the message s/he sends. Or write a message that can only come from given type(s). 

Example 1:  "I never lie" is a message that can come from any type: it is true for both boy knight or girl knave, who always tell truth, and false for both boy knave and girl knight, who always lies. 

Example 2:  "I am a boy" is a message that can only come from a knight. To see this, construct the following truth table:

	boy	girl
knight	T	F
knave	T	F

The message is true for boy knight and false for girl knight, so both types may send it. It is true for boy knave and false for girl knave, so neither type will send it.

Example 3: If a message that can only come from a boy, then its truth values must be as follow:

	boy	girl
knight	T	T
knave	F	F

To exclude the possibility that the message is from a girl, it has to be true for girl knight and false for girl knave.  To make it possible for both types of boys to send this message, it must be true for boy knight and false for boy knave. "I am a knight" is a message that satisfies all these requirements.

Example 4:  Write a message that can only come from a girl knight.  To do so, we first develop the following truth table:

	boy	girl
knight	F	F
knave	T	F

For neither boy knight nor girl knave to send this message, it must be false for both types. For a boy knave not to send this message, it must be true for him. For a girl knight to send this message, it must be false for her.  "I am a boy knave" is a message that meets all these conditions.

Homework:

1. Write a message that can only come from a boy knight.

2. Write a message that can come from anybody but a girl knave.

3. Write a message that cannot come from any of these four types.

4. To refresh your memory about things we learned weeks ago. Suppose that you want to find out if a person you met on the Internet has a brother. The person is of one of the above four types but you don't know which one. You are allowed to send him/her one message to ask a question for a yes/no answer. What question should you ask?

Conclusion on the ``Funny Face'' Problem (homework problem, referring to my last post for  the problem statement):

We can fill out the truth table as follows:

actual case	D said "I did it"		D said "I didn't do it"
	R: D said it	D: I didn't say it	B: D said it	D: I said it
D only said "I did it"	T	F	F	F
D only said "I didn't do it"	F	T	T	T
D said nothing	F	T	F	F

Recall that in the previous case, if D denies both that she said "I did it" and that she said "I didn't do it". The only possibility is that D said neither of these things, so both B and R are knaves.  As a result, the judge has no information to determine whether D is guilty of drawing the funny face.

Now suppose that D confirms to the judge that she said "I didn't do it". As is shown by the truth table above, since D is either a knight or a knave, only two situations are possible:  1) D is a knave who lied both times (she actually said "I did it" and didn't say "I didn't do it"), and 2) D is a knight who told the truth both times (she didn't say "I did it" and said "I didn't do it").  We don't know which case is true, but in either one, D is innocent of drawing the funny face. In the first case, D is a knave, so the fact that she said "I did it" means that she didn't do it. In the second case, D is a knight, so what she said ("I didn't do it") must be true.

Therefore, the answer is that D told the judge that B was telling the truth, so she did say that "I didn't do it". D can be a knight or a knave, and the judge determines that she is innocent regardless which is the case.

Birthday Problem 

As we discussed in the class, the problem is stated as follows: C told the world that her birthday is one of the following ten dates:

May		15	16			19
June				17	18
July	14		16
Aug	14	15		17

C then tells A the day only  and B the month only. So neither of them can pinpoint C's birthday. Then the following conversation takes place:

B: I don't know when is  C's birthday, but I bet A does not know either
A: Now I know
B: Now I know too.

Question: When is C's birthday and how did A and B figure that out?

Answer:  From the first statement from B, the month cannot be May or June. Suppose it was May, since B was told only the month, she cannot exclude the possibility that the birthday is May 19, in which case A would know C's birthday given that she is told the day. Similarly, if it was June, then the day could be 18, in which case A would also be able to figure out C's birthday. The fact that B said that A did not know the birthday means that B must be told either July or August.

After hearing B saying that, A can apply the above reasoning to figure out the month is either July or August. The fact that A is able to use this information to determine C's birthday means that the day A is told cannot be 14 (otherwise, she would  not be able to determine if it is July 14 or Aug 14).

At this point we know that the month is July or August, and the day is 15, 16, or 17, so do A and B. If B is told that the month is August, then it can be August 15 or August 17. Since B said "Now I know too", it means that month she is told must be July, and she can immediately determine (since 14 is not an option) the birthday is July 16.

January 21, 2018 - Math Circle 1.0 Logic & Coordinate Plane

Logic

Answer to the last homework question: Since the funny face is drawn by only one girl, there can be three possibilities: 1) D did it; 2) B did it; and 3) neither D nor B, but some other girl did it. Given that M is a knave, what he said (''it is B instead of D who did it'') must be false, so 2) cannot be the case, but 1) or 3) may happen. We can then conclude that B definitely did not do it, but we are not sure about D: she did it in case 1), and did not do it in case 3).

Next Question: K really wants to find out if D is the girl who drew the funny face. So he brings the case to the court run by the girls. Here D is the D(efendant), R and B are witnesses, and M is the judge. A trial took place and we have obtained the following court record:

• R: D once told us she drew the funny face on K's back.
• D: R is lying, I never said that.
• B: D once said she did not draw the funny face.
• M asks D: Is B telling the truth?
• D said either yes or no, but this part of the record is blurred, so we cannot tell which answer she gave to the judge. 
• M: Now I know whether D is guilty or innocent of drawing that funny face. 

D, R, B can be a knight who always tells the truth or a knave who always lies. 

Question: 1) what did D tell the judge? 2) is D innocent or guilty?

Like the analysis of the last week's problem, here we can answer these questions by considering the two possible answers (yes or no) separately and fill out the truth table in each case. 

First, B and D cannot both tell the truth.  For otherwise, D would have said "I drew the funny face" and "I didn't draw the funny face". One of them must be true and the other must be false, which is not possible since D is either a knight who never lies or a knave who never tells truth.

Thus there can be three situations: 1) D only said "I drew the funny face"; 2) D only said " I didn't drew the funny face", and 3) D said nothing. 

If D's answer to judge's question is no, i.e., B did not tell the truth and D never said "I didn't do it". Then we have the following truth table:

actual case	D said "I did it"		D said "I didn't do it"
	R: D said that	D: I didn't	B: D said that	D: I didn't
D only said "I did it"	T	F	F	T
D only said "I didn't do it"	F	T	T	F
D said nothing	F	T	F	T

Since D is either a knight or a knave, the first two cases, in which D told a truth and a lie, are not possible. So both R and B are knaves, and D is a knight and she said nothing before. In this case, the judge has no way to know whether D actually did it.

Homework:  Suppose that the answer from D to the judge M is yes, i.e., D agree with B that she said "I didn't do it'' before. Use a similar truth table to figure out if D is guilty or innocent.

Coordinate Plane

Introduction to two axesWe have been working on the number line. Today we started looking at the coordinate plane

We still have an origin, but now we have X and Y axes.

We name a point with (x, y). These are called the 'cartesian coordinates'.

Each student should be able to draw these on graph paper where one square = one unit

There are four quadrants

     Which quadrant are the following points in? A(1, -3) B(-3, 1), ...

Draw the points (1, 1), (1, -1), (-1, 1), (-1, -1).
What shape does it make? (most thought it made a square ... but it made an "N")

Homework

1. Plot the function y = |x|

Hint: first fill out the table

x	y = |x|
-4	4
-3
-2
-1
0
1
2

2. Plot and label the points
B(-3, 4)
C(-5, 4)
D(-6, 2)
E(-5, -1)
F(-3, -3)
G(0, 6)
H(3, -3)
I(5, -1)
J(6,2)
K(5,4)
L(3,4)
M(2,3)
N(0,1)

Continue with funny faces:  One boy, K, wants to find out which girl drew the funny face on his back. He knows that one and only one of his friends, E or M, is a detective who can help him with that. But K does not know what type of person his friend is: a knight who always tells truth, or a knave who always lies.

The three friends then have the following conversation:

• E: I am a knight and M is a knave 
• K asks E: Are you the detective? 
• E: Yes. 
• K asks M: Are you the detective? 
• M whispers to K's ear, either yes or no, but we cannot hear. 
• K: Now I know who is the detective 

Question: what did M say to K (yes or no?) and who is the detective?

Key observation:  Even though we don't hear directly what M said to K, we know that K can infer from M's reply to know for sure who is the detective.  So we can try out each of the two possible replies from M, yes or no, and see which one leads to a definite answer about who is the detective. We do that by filling out truth tables.

If M answers yes, then the truth table is:

actual case	E said		M said
	E is knight and M is knave	E is detective	M is detective
E is detective	T	T	F
M is detective	F	F	T

There are two possibilities: E is the detective or M is the detective, and they are listed in the first column of the table. The other columns show whether each thing that E and M said is true (T) or false (F). If E is the detective, then everything he said is true, so he is a knight, and M is a knave who told a lie. If M is the detective, then E is definitely a knave and M obviously is a knight and the detective. If M said yes to K, then both cases are possible, so K cannot know who is the detective.

On the other hand, if M answers no, then the truth table is

actual case	E said		M said
	E is knight and M is knave	E is detective	M is detective
E is detective	F	T	T
M is detective	F	F	F

If E is the detective, then both E and M are telling the truth when answering K's question: are you the detective. But E also told a lie when he said M is a knave. This situation cannot happen since E is either a knight who always tells truth or a knave who alway lies. So there is only possibility, which is the second case:  M is the detective, and everything E and M said is false. As a result, K now knows that both his friends are knaves and M is the detective.

Homework problem:  After a careful look,  K realized that the funny face must be drawn by only one girl. He suspected that girl is D. He told M about his suspicion and asked him to investigate, even though he knows M is a knave who always lies. M comes back and tells K that

"It is B instead of D who did it."

Question:

1. Did B draw the funny face (yes, no, maybe)?
2. Did D do it (yes, no, maybe)?

January 21, 2018 - Math Circle 1.0: Absolute Value & Logics

Logic

Here's the summary of our discussions on logic problems in the past two weeks.

Includes solutions to last week's homework and new homework for this week! 

Absolute Value

first, a motivating exercise on the number line:

 

Absolute Value definition

• defined as distance from the origin
• Use $| |$ to indicate absolute value
• if $x>0$ then $|x|=x$
• if $x<0$ then $|x|=-x$
• if $x=0$ then $|x|=0$

Absolute Value examples

First we went through a few questions like find the absolute value of -1, 10, 0, a zillion. Then:

hint: can x = 1? can x = 0? can x = 3? can x = a zillion?

January 7, 2018 - Math Circle 1.0: The number line

The number line

From Gelfand et al 1990 Method of Coordinates:

Introduction to coordinates

The method of coordinates allows us to define where something is
examples
- airplane dropping a package with food in a disaster area
- satellite launch
- cupid --> address
- chess
- lat, lon defines point on earth sfc

Can convert points / lines into numbers

Dimensions: line = 1, plane = 2

The number line

1. origin
2. positive direction
3. unit of measurement

A(5) located on a number line

Defining a point
- ea. point defined by a single number
- the coordinate 5 is 5 units from origin
- notation: A(5) "point A is at coordinate +5"

Correspondence:
- ea. point on line has one and only one number coordinate
- ea. coordinate has one and only one location on the line

Draw A(5) and B(5)
Draw A(2)
    - hint: does not exist

Exercises

draw number line from -5 to 5
place A(5), B(-4), C(1½) on line

- place M(2); which two coordinates are 2 units from M(2)?

Exercise as written in a student's notebook

.

- if B(b) is to the right of A(a), which is greater (a or b)

- Without using a number line, which of the following points is to the right of the other?

•  A(-3), B(-4)
• A(3), B(4)

• A(a), B(-a)

For A(a), B(-a): there are three conditions to consider (a=0, a>0, a<0)

if a=0, A and B are at same point
if a> 0 ...
if a < 0 ...

Homework 

Which of the following is greater:

• M(x) or N(2x)
• A(c) or B(c+2)
• A(x) or B(x²)