The Puzzle Thread

[Hoodat]

Feel free to post your own.

Here's one for starters.

The Counterfeit Coin.

You have twelve coins, each marked 1 through 12.  Eleven of the coins are made with a precious metal.   The remaining coin is counterfeit.  By visual inspection, all twelve of the coins are indistinguishable in shape, size, and color.  The eleven genuine coins share the same weight.  But the counterfeit coin is either heavier or lighter than the other eleven.



The only tool you have at your disposal is a balance scale.  You can put as many or as few coins on the scale at any time.

Q:  What is the fewest number of weightings required to successfully determine which coin is the counterfeit and whether it is heavier or lighter than the other eleven?

[EdinVA]

3

[Hoodat]

Three is the correct answer.  Here's the math.  Twelve coins translates to 24 possible outcomes - Coin 1 is heavier, Coin 1 is lighter, Coin 2 is heavier, Coin 2 is lighter, . . .  Using the balance, there are three possible outcomes - Tilts to the left, balances, tilts to the right.  The lowest power of 3 that exceeds 24 is 3³ = 27.  So three weightings should cover it.

The next thing is to ensure that no two ball have the same weigh pattern.
Here are the weigh patterns for each ball.  The first denotes right scale for weighing 1, left for 2, right for 3.

•   R | L | R
•   R | R | L
•   L | R | R
•   L | L | -
•   - | L | L
•   R | - | R
•   L | R | -
•   - | R | L
•   R | - | L
•   - | - | R
•   - | L | -
•   L | - | -

Results:

Left down, right down, left down = Coin 1 is heavier
Right down, left down, right down = Coin 1 is lighter
Left down, left down, right down = Coin 2 is lighter
Right down, right down, left down = Coin 2 is heavier
.  .  .  . 
Left heavier, even, left heavier = Coin 6 is lighter
Right heavier, even right heavier = Coin 6 is heavier
.  .  .  . 
Even, even, right is heavier = Coin 10 is heavier.
.  .  .  .

And so on.

[Hoodat]

Since SARS-CoV-2 is taking up all the news these days, here is one on disease probability.

Let's say there is a disease out there that only 1% of the population has.  There is a test for this disease that is 99% accurate (meaning that 1% of tests give false results).
Now, consider a person who takes the test and tests positive for the disease.  What is the probability that this person actually has the disease?

[EdinVA]

99%

[Axeslinger]

@EdinVA  @Hoodat

3 is incorrect.  You don’t know if the bad coin is lighter or heavier.  So your first weighing you split the twelve into two stacks of 6 and weigh them against each other. One side will be heavier.   But you have to reweigh BOTH of those stacks to determine if you’re looking for a lighter or a heavier coin.    When you take those two stacks of six and separate them into 3 each, one of the groups of 6 will balance the other will not.  At that point you know if you’re looking for a lighter or heavier coin.  Thus, the answer for MINIMUM weighings is 4

[Hoodat]

@Axeslinger

Weigh the coins like this.

First weigh

 LEFT                RIGHT
3-4-7-12 ---------- 1-2-6-9


Second weigh

 LEFT                RIGHT
1-4-5-11 ---------- 2-3-7-8


Third weigh

 LEFT                RIGHT
2-5-8-9 ---------- 1-3-6-10

Consider these outcomes for the three trials

Lean LEFT   |  Lean RIGHT |  Lean LEFT   =  Coin 1 is Light
Lean RIGHT  |  Lean LEFT  |  Lean RIGHT  =  Coin 1 is Heavy
Lean LEFT   |  Lean LEFT  |  Lean RIGHT  =  Coin 2 is Light
Lean RIGHT  |  Lean RIGHT |  Lean LEFT   =  Coin 2 is Heavy
Lean RIGHT  |  Lean LEFT  |  Lean LEFT   =  Coin 3 is Light
Lean LEFT   |  Lean RIGHT |  Lean RIGHT  =  Coin 3 is Heavy
Lean RIGHT  |  Lean RIGHT |   --EVEN--   =  Coin 4 is Light
Lean LEFT   |  Lean LEFT  |   --EVEN--   =  Coin 4 is Heavy
 --EVEN--   |  Lean RIGHT |  Lean RIGHT  =  Coin 5 is Light

And so on.

[Hoodat]

99%

@EdinVA

99% is incorrect.

[EdinVA]

@EdinVA

99% is incorrect.

@Hoodat

Let's say there is a disease out there that only 1% of the population has.  There is a test for this disease that is 99% accurate (meaning that 1% of tests give false results).
Now, consider a person who takes the test and tests positive for the disease.  What is the probability that this person actually has the disease?

The problem identified the test results as positive, therefore, it is a binary question relating to the accuracy of the test only.

Had the test results not been included, then the question would have been more complex.
 

[240B]

What if I flip a penny 100 times and 100 times it lands on Heads
If I flip it one more time, what are the odds that it will land on tails?

[EdinVA]

What if I flip a penny 100 times and 100 times it lands on Heads
If I flip it one more time, what are the odds that it will land on tails?

50/50...

[240B]

50/50...

@EdinVA

That is the correct/incorrect answer.
The penny is obviously defective or weighted somehow.
It will land on heads.

[EdinVA]

@EdinVA

That is the correct/incorrect answer.
The penny is obviously defective or weighted somehow.
It will land on heads.

    You're right... forgot about that part...

[Hoodat]

That is the correct/incorrect answer.
The penny is obviously defective or weighted somehow.
It will land on heads.

Nice.

[Hoodat]

@Hoodat

Let's say there is a disease out there that only 1% of the population has.  There is a test for this disease that is 99% accurate (meaning that 1% of tests give false results).
Now, consider a person who takes the test and tests positive for the disease.  What is the probability that this person actually has the disease?

The problem identified the test results as positive, therefore, it is a binary question relating to the accuracy of the test only.

Had the test results not been included, then the question would have been more complex.
 

@EdinVA

Had the test result not been included, then the question would be, what is the probability that someone has the disease.  That number was already given - 1%.  But the result was included, making this a conditional question in that a person tested positive.  This person testing positive either has or does not have the disease.  What is the probability that he does?

[Hoodat]

Anybody else want to take a guess at this before I post the answer?

Let's say there is a disease out there that only 1% of the population has.  There is a test for this disease that is 99% accurate (meaning that 1% of tests give false results).
Now, consider a person who takes the test and tests positive for the disease.  What is the probability that this person actually has the disease?

[Hoodat]

Answer:  The probability is 50%.

Bayes Theorem states   

The probability of actually having the disease if testing positive is (.99 * .01) / [(.99 * .01) + (.01 * .99)] = 0.5

Consider a population of 10,000.  Of that population, 100 people have the disease, and 9,900 do not.
Of the 100 that have the disease, 99 will test positive while 1 tests negative.
Of the 9,900 that do not have it, 99 will test positive while 9,801 test negative.
This leaves 198 people (out of 10,000) who test positive.  Of the 198, 99 have the disease, and 99 do not.  50%.

Keep that in mind if you take a SARS-2 test and it comes up positive.

[EdinVA]

Answer:  The probability is 50%.

Bayes Theorem states   

The probability of actually having the disease if testing positive is (.99 * .01) / [(.99 * .01) + (.01 * .99)] = 0.5

Consider a population of 10,000.  Of that population, 100 people have the disease, and 9,900 do not.
Of the 100 that have the disease, 99 will test positive while 1 tests negative.
Of the 9,900 that do not have it, 99 will test positive while 9,801 test negative.
This leaves 198 people (out of 10,000) who test positive.  Of the 198, 99 have the disease, and 99 do not.  50%.

Keep that in mind if you take a SARS-2 test and it comes up positive.

geezz @Hoodat .. I would have to study for 6 months to pass that test....

[Hoodat]

A man had purchased his dream sports car and decided to take his 10-year-old son for a drive.  As they were driving down the highway at high speed, a nail in the roadway punctured the tire causing the man to lose control.  A terrible accident ensued as the car swerve off the road and crashed into a tree.  The man was instantly killed upon impact.  His son was still alive, but had severe injuries.  An ambulance picked up the boy and rushed him to the hospital.  They immediately wheeled him into the ER for emergency surgery.  The doctor walks in, looks at the boy, and says, "I cannot operate on this boy.  He is my son."

How can this be?

[EdinVA]

A man had purchased his dream sports car and decided to take his 10-year-old son for a drive.  As they were driving down the highway at high speed, a nail in the roadway punctured the tire causing the man to lose control.  A terrible accident ensued as the car swerve off the road and crashed into a tree.  The man was instantly killed upon impact.  His son was still alive, but had severe injuries.  An ambulance picked up the boy and rushed him to the hospital.  They immediately wheeled him into the ER for emergency surgery.  The doctor walks in, looks at the boy, and says, "I cannot operate on this boy.  He is my son."

How can this be?

The doctor bought the car and had a friend drive it.

[EdinVA]

There are 2 mothers, 2 daughters, 1 grandmother and 1 granddaughter. How many people are there?

[Right_in_Virginia]

There are 2 mothers, 2 daughters, 1 grandmother and 1 granddaughter. How many people are there?

Grandma, her daughter and granddaughter = 3

[Hoodat]

The doctor bought the car and had a friend drive it.

@EdinVA

No.  The man who bought the car was the driver.  The only occupants of the car were the man and his son.