Easy math puzzles

Quote: Gialmere

Correct!!

He lives.
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Quote: ThatDonGuy

I didn't want to answer this one, as I have seen it before - a variation is in Smullyan's The Riddle of Scheherazade:

Answer

You will not keep both bills, nor will you pay me $100.

If the statement is true, then the student keeps both bills - but that makes the statement false. Therefore, the statement must be false.
This means that the student did keep both bills, or paid the professor $100. However, since the statement is false, the student had to give the professor one of the bills, so the only way the statement can be false is if the student pays the $100.

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Quote: Gialmere

It's toughie Tuesday. Let's go a little bit country...

A farmer has three daughters. He is getting old and decides to split his 1-mile-by-1-mile farm equally among his daughters using fencing.

What is the shortest length of fence he needs to divide his square farm into three sections of equal area?
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I don't claim any of these is optimal.

My first thought was the following type of borders.



I find the fencing required here is 5/3.

Next, I thought of these diagonal fences:



Here, I get an answer of sqrt(3)=~ 1.7321, which is worse than the first idea.

Next, I went with fencing like this:



I'll just say, for now, the total fencing with the optimal measurements is 1.634912503, which is better than 5/3. If this deemed correct, I'll provide my solution.

Quote: Wizard

Here is a variant I've heard of the Three Daughters problem.

A farmer has a square piece of land. Somewhere under his land, an underground straight pipe crosses it. The farmer is tasked with finding this pipe. He decides to dig ditches until he finds it. What is the least amount of ditch length that will guarantee finding it?
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Quote: unJon

Does “crosses it” mean the straight pipe intersects any two sides of the square? Or must it intersect to parellel sides?
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Quote: Wizard

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I don't claim any of these is optimal.

My first thought was the following type of borders.



I find the fencing required here is 5/3.

Next, I thought of these diagonal fences:



Here, I get an answer of sqrt(3)=~ 1.7321, which is worse than the first idea.

Next, I went with fencing like this:



I'll just say, for now, the total fencing with the optimal measurements is 1.634912503, which is better than 5/3. If this deemed correct, I'll provide my solution.

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Quote: charliepatrick

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There's an associated problem of joining three towns with roads; it transpires that roads need to meet at 120o to each other.

Nice idea - but WRONG ANSWER - but leads to the second idea!
Applying the same logic here I'm guessing it's similar to the "T" but then you alter the top of the T so the lines are at 120o. I haven't checked the maths but let's assume the height to the junction is y, and the points on the edge are x further up.

The area for each of the lower parts is y/2+x/4 = 1/3. The area for the top is 1-y-x/2. Also you know the Hypotenuse of the "triangle" is 2x, so the base is SQRT(3*)x. I think y=2/3-1/(4*SQRT(3) and x=1/(2*SQRT(3))

The total length of fences is y+4x = 2/3 + SQRT(3)*(2/3-1/12) = 2/3 + SQRT(3)*7/12. = 1.677.

SECOND IDEA
However this leads to considering a similar idea, draw a line up the middle and then create the triangles to give each person an area of 1/3. For instance if y=.6 then x=.133333, this isn't a 90/60/30 so the fence isn't 2x, in fact the total length of the fences is 1.6349. It transpires y=0.6022 gives 1.634913. I don't know why this isn't the 120o solution above or an elegant mathematical expression for it.

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Quote: Gialmere

The Y fence configuration earns you both a half point. There is, however, room for tweaking.
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Quote: Wizard

Quote: Gialmere

The Y fence configuration earns you both a half point. There is, however, room for tweaking.
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I think I'm going to have to settle upon the half point. However, I'm interested in a hint or the outright answer.
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Quote: Gialmere

The Y fence configuration earns you both a half point. There is, however, room for tweaking.
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Quote: charliepatrick

1.632238

My arc method, via a spreadsheet, gets 1.63223813 using Y=.530196478. You can work out the hypotenuse and hence the angle, this gives the length of the arc and the area of the slice.

Sadly it's by trial and error, so I haven't got a formula for the lengths.
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Quote: Gialmere

Quote: charliepatrick

1.632238

My arc method, via a spreadsheet, gets 1.63223813 using Y=.530196478. You can work out the hypotenuse and hence the angle, this gives the length of the arc and the area of the slice.

Sadly it's by trial and error, so I haven't got a formula for the lengths.
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You're hot on the trail but missing something. You can still squeeze a little more juice (i.e. a little less fence) out of it.
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Quote: Wizard

Here is a variant I've heard of the Three Daughters problem.

A farmer has a square piece of land. Somewhere under his land, an underground straight pipe crosses it. The farmer is tasked with finding this pipe. He decides to dig ditches until he finds it. What is the least amount of ditch length that will guarantee finding it?
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In order to be sure to find the intersecting pipe, trenches need to be dug that block each side from each other side. So digging a trench around the perimeter works but has a length of 4. You can save by digging around just three sides for a length of 3. Better still you can start in each corner and make two trenches along the diagonal for a length of 2 * sqrt(2) = 2.828427

I think you could do best though by modifying the X diagonals by having a straight line in the middle connecting the left half > and right half < (or top and bottom). So think of four diagonal lines connecting two halves of the X. Like this:



As the length of the middle connecting line (call it X) increases, each of the four diagonal lines (call it Y) shrinks.

The length of the trench is L = X + 4 Y.

The formula between X and Y is based on the Pythagorean formula.

f(X) = Y = sqrt( 1/2 ^ 2 + (1/2 - X/2)^2 )
Simplify to Y = sqrt(1/2 - X/2 + (X^2)/4)

Substitute in for Y in the length formula.

L = X + 4 * sqrt(1/2 - X/2 + (X^2)/4)

Take the derivative to find the minimum of L. (I used Wolfram Alpha) to get the minimum at X = 1 - 1/sqrt(3).

With L = 1 + sqrt(3) = 2.732051

Quote: unJon

I cannot think of a better answer than this:

Show Spoiler

In order to be sure to find the intersecting pipe, trenches need to be dug that block each side from each other side. So digging a trench around the perimeter works but has a length of 4. You can save by digging around just three sides for a length of 3. Better still you can start in each corner and make two trenches along the diagonal for a length of 2 * sqrt(2) = 2.828427

I think you could do best though by modifying the X diagonals by having a straight line in the middle connecting the left half > and right half < (or top and bottom). So think of four diagonal lines connecting two halves of the X. Like this:



As the length of the middle connecting line (call it X) increases, each of the four diagonal lines (call it Y) shrinks.

The length of the trench is L = X + 4 Y.

The formula between X and Y is based on the Pythagorean formula.

f(X) = Y = sqrt( 1/2 ^ 2 + (1/2 - X/2)^2 )
Simplify to Y = sqrt(1/2 - X/2 + (X^2)/4)

Substitute in for Y in the length formula.

L = X + 4 * sqrt(1/2 - X/2 + (X^2)/4)

Take the derivative to find the minimum of L. (I used Wolfram Alpha) to get the minimum at X = 1 - 1/sqrt(3).

With L = 1 + sqrt(3) = 2.732051

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Quote: Wizard

Here is a variant I've heard of the Three Daughters problem.

A farmer has a square piece of land. Somewhere under his land, an underground straight pipe crosses it. The farmer is tasked with finding this pipe. He decides to dig ditches until he finds it. What is the least amount of ditch length that will guarantee finding it?
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Consider the square ABCD and the middle line (EFG) As has been shown (you have to cover all the corners and create lines) the objective is to join ABCD with roads. Without going into the proof it transpires when connecting three points with the shortest length the roads have to meet at 120o. Therefore FBG (and the other similar triangles) are 30/60/90, hence FB = SQRT(3)/3. The total roads are four lines similar to FB and EF, which is 1 - 2*FG. This is 4*[SQRT(3)/3]+[1-2*SQRT(3)/6] = 1 + SQRT(3).

Quote: charliepatrick

Show Spoiler

Consider the square ABCD and the middle line (EFG) As has been shown (you have to cover all the corners and create lines) the objective is to join ABCD with roads. Without going into the proof it transpires when connecting three points with the shortest length the roads have to meet at 120o. Therefore FBG (and the other similar triangles) are 30/60/90, hence FB = SQRT(3)/3. The total roads are four lines similar to FB and EF, which is 1 - 2*FG. This is 4*[SQRT(3)/3]+[1-2*SQRT(3)/6] = 1 + SQRT(3).

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Quote: Gialmere

It's toughie Tuesday. Let's go a little bit country...



A farmer has three daughters. He is getting old and decides to split his 1-mile-by-1-mile farm equally among his daughters using fencing.

What is the shortest length of fence he needs to divide his square farm into three sections of equal area?

Part #1

As you begin to plan your fence-building, you might sketch out a few blueprints and measure how much fence you’d need.



You start off simple, just planning to build two plain old east-to-west fences across the farm. These divide your farm into three identically shaped horizontal slices, giving you the equal areas you desire. This first arrangement uses 2 miles of fence.

Part #2

But nothing is forcing you to go east to west. So you draw a blueprint where a vertical fence line meets a horizontal fence line, as shown in the second diagram. Again, you’ve divided your farm into three regions of equal area, although this time not identically shaped. This arrangement uses 1+2/3 or about 1.67 miles of fence. Much better! If you made it this far, you are a conscientious and efficient farmer and have done the agricultural sector of Riddler Nation proud.

Part #3

But then you have an epiphany: There’s no reason the fences need to be east-west or north-south at all — they could be diagonal! So you sketch out a blueprint like the third diagram, where fences meet in a Y shape to split up the farm. But now things get a little more complicated because you need to figure out where exactly to put the diagonal lines and how much fence this’ll take. You’ll need a little algebra and a touch of calculus. First you figure out what the distances x and y must be.2 You know each region must have an area of 1/3, so you know, using the formulas for the area of a rectangle (base times height) and a triangle (one-half base times height), that y/2+x/4= 1/3. In other words, y=2/3−x/2

So how much fence does this arrangement use? It uses one vertical piece (length 2/3−x/2
) and two diagonal pieces. The diagonal pieces are the hypotenuses of right triangles, so we can get their length from the Pythagorean theorem: x2+(1/2)2= that length of fence squared. Then we take the square root of that (which is the length of one diagonal) and add two of them to y, which gets us the total length of fence in this arrangement. (Call it L.)



Your job as super-efficient farmer is to select the x that minimizes that total length L. You can do this by taking the derivative of L with respect to x and setting it equal to zero. That gives x=1/(2√15). Plugging that back into our equation for L, this arrangement uses about 1.635 miles of fence. Even better than our second fence! If you made it this far, you earned a blue ribbon in the Riddler Nation agricultural expo.

Part #4

But then you have another epiphany and nearly pass out: There’s no reason the fences need to be straight — they could be curved! So you sketch a blueprint like the fourth diagram. It’s similar to the third, except the “arms” of your fence arrangement are bits of larger circles.

The math here gets more complicated still. But in the end it turns out that z, the “stem” of your fence arrangement, is about 0.576 miles long, the circle itself has radius one, the angle pictured is 30 degrees, and the entire thing takes about 1.623 miles of fence. Our best result yet! If you made it this far, congratulations: You’re Riddler Nation’s new minister of agriculture.

Quote: Gialmere

You're hot on the trail but missing something. You can still squeeze a little more juice (i.e. a little less fence) out of it.
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1.623278

Please see enclosed picture. Essentially construct a circle of unit radius - point O is the centre just off the picture (I've included another unit circle CB, so just slid it down to point E). This way the angle of FEG is 30 degrees, thus ensuring the angles of the lines meet at 120 degrees (which is what I was hoping for).

There is a mathematical expression (if you go through the steps you could work it out) for this number.

Quote: charliepatrick

Quote: Gialmere

You're hot on the trail but missing something. You can still squeeze a little more juice (i.e. a little less fence) out of it.
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1.623278

Please see enclosed picture. Essentially construct a circle of unit radius - point O is the centre just off the picture (I've included another unit circle CB, so just slid it down to point E). This way the angle of FEG is 30 degrees, thus ensuring the angles of the lines meet at 120 degrees (which is what I was hoping for).

There is a mathematical expression (if you go through the steps you could work it out) for this number.

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joke

Quote: Gialmere

Quote: Gialmere

It's toughie Tuesday. Let's go a little bit country...



A farmer has three daughters. He is getting old and decides to split his 1-mile-by-1-mile farm equally among his daughters using fencing.

What is the shortest length of fence he needs to divide his square farm into three sections of equal area?

Buzzz

Semi-solve by Wizard and charliepatrick with a correct guess from unJon.

Here is the official solve at Riddler...

Part #1

As you begin to plan your fence-building, you might sketch out a few blueprints and measure how much fence you’d need.



You start off simple, just planning to build two plain old east-to-west fences across the farm. These divide your farm into three identically shaped horizontal slices, giving you the equal areas you desire. This first arrangement uses 2 miles of fence.

Part #2

But nothing is forcing you to go east to west. So you draw a blueprint where a vertical fence line meets a horizontal fence line, as shown in the second diagram. Again, you’ve divided your farm into three regions of equal area, although this time not identically shaped. This arrangement uses 1+2/3 or about 1.67 miles of fence. Much better! If you made it this far, you are a conscientious and efficient farmer and have done the agricultural sector of Riddler Nation proud.

Part #3

But then you have an epiphany: There’s no reason the fences need to be east-west or north-south at all — they could be diagonal! So you sketch out a blueprint like the third diagram, where fences meet in a Y shape to split up the farm. But now things get a little more complicated because you need to figure out where exactly to put the diagonal lines and how much fence this’ll take. You’ll need a little algebra and a touch of calculus. First you figure out what the distances x and y must be.2 You know each region must have an area of 1/3, so you know, using the formulas for the area of a rectangle (base times height) and a triangle (one-half base times height), that y/2+x/4= 1/3. In other words, y=2/3−x/2

So how much fence does this arrangement use? It uses one vertical piece (length 2/3−x/2
) and two diagonal pieces. The diagonal pieces are the hypotenuses of right triangles, so we can get their length from the Pythagorean theorem: x2+(1/2)2= that length of fence squared. Then we take the square root of that (which is the length of one diagonal) and add two of them to y, which gets us the total length of fence in this arrangement. (Call it L.)



Your job as super-efficient farmer is to select the x that minimizes that total length L. You can do this by taking the derivative of L with respect to x and setting it equal to zero. That gives x=1/(2√15). Plugging that back into our equation for L, this arrangement uses about 1.635 miles of fence. Even better than our second fence! If you made it this far, you earned a blue ribbon in the Riddler Nation agricultural expo.

Part #4

But then you have another epiphany and nearly pass out: There’s no reason the fences need to be straight — they could be curved! So you sketch a blueprint like the fourth diagram. It’s similar to the third, except the “arms” of your fence arrangement are bits of larger circles.

The math here gets more complicated still. But in the end it turns out that z, the “stem” of your fence arrangement, is about 0.576 miles long, the circle itself has radius one, the angle pictured is 30 degrees, and the entire thing takes about 1.623 miles of fence. Our best result yet! If you made it this far, congratulations: You’re Riddler Nation’s new minister of agriculture.

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Quote: unJon

Hmmm. I’m too lazy to calculate it, but I do wonder if the below two arcs would be even better. Both red lines are arcs of a unit circle.


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Quote: Gialmere

You have two faucets for your shower, one that releases hot water and one that releases cold water, but at varying pressures.

If the hot water is 105 degrees Fahrenheit and the cold water is 45 degrees Fahrenheit, and you could rotate the faucet at most 360 degrees, how much would you have to rotate the cold water faucet to get the temperature to be 98.6 degrees Fahrenheit, if the hot water is turned on all the way, and could not be turned down?
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Quote: ChesterDog

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Turn the cold water faucet on by an angle of 43 degrees.

Call the hot water flow H. With respect to the cold temperature, the energy supplied to the shower is proportional to (H)(105 - 45).

The cold water flow is some fraction (f) of the hot water flow. With respect to the cold water temperature, the energy supplied to the shower by the cold water is proportional to (fH)(45 - 45) = 0.

The total flow of 98.6F water is H + fH. The energy of the 98.6F water with respect to 45F is (H + fH)(98.6 - 45).
Then: (105 - 45)H = (98.6 - 45)( H + fH).

Solve for f: f = 8 / 67.

Then the angle for the cold water would be: 360(8/67) ≈ 43 degrees.

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Quote: Gialmere

Use each digit 1-9 only once each, so that one number multiplied by another equals the third. One example is 12*483=5796.

How many more of these can you find?
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Quote: Gialmere

Use each digit 1-9 only once each, so that one number multiplied by another equals the third. One example is 12*483=5796.

How many more of these can you find?

Quote: Gialmere

Use each digit 1-9 only once each, so that one number multiplied by another equals the third. One example is 12*483=5796.

How many more of these can you find?
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• The first digit of the number is the number of 0's in the entire number.
• The second digit of the number is the number of 1's in the entire number.
• The third digit of the number is the number of 2's in the entire number.
• The 4th digit of the number is the number of 3's in the entire number.
• The 5th digit of the number is the number of 4's in the entire number.
• The 6th digit of the number is the number of 5's in the entire number.
• The 7th digit of the number is the number of 6's in the entire number.
• The 8th digit of the number is the number of 7's in the entire number.
• The 9th digit of the number is the number of 8's in the entire number.
• The 10th digit of the number is the number of 9's in the entire number.

Quote: Wizard

Find a ten-digit number such that:

• The first digit of the number is the number of 0's in the entire number.
• The second digit of the number is the number of 1's in the entire number.
• The third digit of the number is the number of 2's in the entire number.
• The 4th digit of the number is the number of 3's in the entire number.
• The 5th digit of the number is the number of 4's in the entire number.
• The 6th digit of the number is the number of 5's in the entire number.
• The 7th digit of the number is the number of 6's in the entire number.
• The 8th digit of the number is the number of 7's in the entire number.
• The 9th digit of the number is the number of 8's in the entire number.
• The 10th digit of the number is the number of 9's in the entire number.

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Quote: ThatDonGuy

Answer

7,210,000,100
Seven 0s, two 1s, one 2, and one 7

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Quote: Wizard

Quote: ThatDonGuy

Answer

7,210,000,100
Seven 0s, two 1s, one 2, and one 7

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Reply

I count only six 0's

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Quote: Wizard

Find a ten-digit number such that:

• The first digit of the number is the number of 0's in the entire number.
• The second digit of the number is the number of 1's in the entire number.
• The third digit of the number is the number of 2's in the entire number.
• The 4th digit of the number is the number of 3's in the entire number.
• The 5th digit of the number is the number of 4's in the entire number.
• The 6th digit of the number is the number of 5's in the entire number.
• The 7th digit of the number is the number of 6's in the entire number.
• The 8th digit of the number is the number of 7's in the entire number.
• The 9th digit of the number is the number of 8's in the entire number.
• The 10th digit of the number is the number of 9's in the entire number.

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Quote: ThatDonGuy

In that case...

It's 6,210,001,000
Six 0s, two 1s, one 2, and one 6

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Quote: unJon

Show Spoiler

Let’s do it iteratively from a number that’s all 0s.

Start with 0,000,000,000.
Go to: 9,000,000,000.
Next: 8,000,000,010.
Next: 6,210,001,000.
Done.

Is that the only one?

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Quote: Wizard

I agree! That is the same answer I came up with. I do not know if there are other answers.
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Quote: Gialmere

It's easy Monday. Let's play poker...



During a game of five card draw poker, played with a standard deck, you are dealt a hand with the following characteristics:

--It contains no aces or face cards.
--No two cards have the same value.
--All four suits are present.
--The total value of the odd cards equals the total value of the even cards.
--There are no three card straights.
--The total value of the black cards is 10.
--The total value of the hearts is 14.
--The card with the lowest value is a spade.

Exactly what are the five cards in your hand?
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Quote: Gialmere

It's easy Monday....Exactly what are the five cards in your hand?...

Quote: rsactuary

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I think 9 hearts, 5 hearts, 4 diamonds, 8 clubs and 2 spades satisfies the requirements

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Quote: charliepatrick

you can use logic to get there

Black=10, Hearts=14, these both use up two cards so the diamond is a 4d. The Blacks or Hearts are either both odd or both even. Because of the 4d, the Blacks also have to be even; and the total of even=odd=14.
The Hearts have to be 9h and 5h to get to 14. The Blacks, so they can't form a straight or be a 4, are 2 and 8, with the lowest being 2s, gives 8c .

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Quote: ThatDonGuy

Quote: Wizard

I agree! That is the same answer I came up with. I do not know if there are other answers.
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There "should be" a way to prove this without using brute force, but a brute force search shows that this is the only 10-digit solution.

A few notes, if anyone is trying to do an analytical proof:
"Obviously," the sum of the digits is 10
The digits in the 5-9 positions must be 0 or 1 (as a 2 would require more than 10 digits)
The first digit (the number of 0s) cannot be 0, as then there would be one 0 in the number
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