#### Logic

- If A is married and looks at B and B looks at C who is unmarried, is a married person looking at an unmarried person?

yes

no

can’t tell

## Solution*
Yes*

*Whether B is married or unmarried a married person is looking at an unmarried person*

- You have 3 coins. 1 weighs less than the rest. Can you find it with just 1 weighing?
- Divide the three coins into two groups of one coin each, and leave the third coin aside.
- Place one coin on each side of the balance scale.
**If the scale balances**, then neither of the two coins on the scale is the lighter one, meaning the coin you left aside is the lighter coin.**If the scale does not balance**, the side that goes up (the lighter side) contains the lighter coin.

## Solution

To find the lighter coin among the three coins with just one weighing, you can use a balance scale and follow these steps:

There are two possible outcomes from this weighing:

This method ensures you can identify the lighter coin among the three with just one use of the scale.

- 10 red socks, 10 blue socks. How many to pull for matching pair? How many to pull for 1 of each?

## Solution

3 and 11 respectively

- If you have 3 you have 3, if you have 2 you have 2, if you have 1 you have none. What is it?

## Solution

Choice

#### Arithmetic

- Dinner cost $60 including a 20% tip. How much was the meal?

## Solution

1.2x = 60

x = 50

- I'm 3x your age today. In 3 years I'll be twice your age. What are our ages?

## Solution

y = 3x

y + 3 = 2(x + 3)

3 and 9

- A bat costs $1 more than a ball. Together they cost $1.10. How much do they each cost?

## Solution

ball = .05, bat = 1.05

- What's bigger the square of an average or the average of the squares?

## Solution

average of the squares

- Cigarette question: A miser can make 1 cigarette from 5 cigarette butts. What is the maximum number of cigarettes the miser can make from 1000 cigarette butts ?

## Solution

he first makes 1000/5 = 200 cigarettes which gives him 200 butts in the end
then he has 200 butts to make 200/5= 40 cigarettes which gives 40 butts
again he has 40/5 = 8 cigarettes which gives 8 butts
8/5 = 1.6 but he makes only 1 cigarette of that.
3 butts remain in the end - useless (as he cant make a cigarette with 3 butts)
Total cigarettes made = 200 + 40 + 8 +1 = 249 cigarettes

- You drive to the store and back. The store is 50 miles away. You drive 50mph to the store and 100mph coming back. What’s your average MPH for the trip?

## Solution

66.66 mph

(1 hour going and 30 minutes returning)

- You drive 1 mile at 30mph, how fast do you have to drive the second mile to average 60 mph for the trip?

## Solution

Can't be done. The first mile took 2 and to drive 2 miles at 60mph you must complete the trip in 2 min

- Solve without a calculator:

## Solution

We can then realize that:

So we can substitute as our target instead of just :

The target is simply

= 243

**So in the original equation x =**

**243 or -243**#### Probability & Stats

- If 10% of population is left handed and 1% of women are left handed, what percentage of men are left handed? (assume equal amounts of men and women in the population)

## Solution

.5 women are left handed

9.5/10 left handed people must be men

9.5/50 = 19% of men are left handed

- Expected Value
- coin toss game
- Insurance on a iPad
- Roulette wheel
- 2 pointer vs 3 pointer
- Getting a blackjack from a full deck

## EV of:

- Flip 100 coins, labeled 1 through 100. Alice checks the coins in order (1, 2, 3, …) while Bob checks the odd-labeled coins, then the even-labeled ones (so 1, 3, 5, …, 99, 2, 4, 6, …). Who is more likely to see two heads
*first*? (via twitter)

## Solution

*The first coin is irrelevant.*

*The second coin they have an equal chance of finding heads.*

*But on the 3rd look Alice is looking at a coin that we know didn't stop the game so it's a wasted look*

**Divisibility Rules**

**Divisibility Rules**

- Divisible by 10 if ends in 0

- Divisible by 2 if even

- Divisible by 3 if digital roots is 3

- Divisible by 4 if the last 2 digits of a number are divisible by 4

- Divisible by 5 if the last digit is divisible by 0 or 5

- Divisible by 6 if it's divisible by 3 and even

- Divisible by 7 if the difference between twice the unit digit of the given number and the remaining part of the given number should be a multiple of 7 or it should be equal to
**0**

- Divisible by 8 if the last 3 numbers are 000 or last 3 are divisible by 8

- divisible by 9 if digital roots is 9

#### Misc Topics

- Richter scale as log scale. A 6 is 1000x a 3 not 2x

- Socratic lesson: Teaching about mean absolute deviation (easier than variance)
**MAD and daylight savings time***Daylight savings time causes accidents but how do we know?*

*If CA averages 100 car accidents a day, then would you be surprised if there was 92? 105? 300?*

*There's an implicit level of “surprise”*

*Compute MAD by averaging the distances of each day’s accidents from 100. The more a sample exceeds the average the more likely there result is not random*

**Triangle method for summing the numbers from 1 to N**

**Passphrases > passwords**

Passphrases: hard for computer to crack (compute time for long strings) and easy for human to remember vs

password: easy for computer to crack hard for human to remember

- Sum of finite geometric series
- derivation
- socratically show terms in the sequence
- set equal to the same sequence times
*r* - solve for Sn
**Depreciation**: When an asset depreciates by a constant percentage each year, the total value lost over a number of years can be calculated as the sum of a geometric series.**Loan Payments**: The formula for calculating the remaining balance of a loan involves a geometric series when the loan is being repaid in equal installments over time.**Investment Growth**: The total value of an investment over time that receives a constant rate of interest compounded annually can be modeled as a geometric series.**Electronics**: In a resistor-capacitor (RC) circuit, the voltage across the capacitor or the charging and discharging of the capacitor can be described by a geometric series.**Computer Science**: In algorithms, particularly divide and conquer algorithms like the binary search, the time complexity can often be expressed as a geometric series.**Physics**: In certain problems in physics, such as calculating the total distance covered by a bouncing ball where each bounce is a fraction of the height of the previous bounce.**Acoustics**: Echoes can be thought of as a geometric series, where each successive echo has a smaller amplitude than the previous one

*applications *

Real-life examples of summing a finite geometric series include: