Grad Wiki - User contributions [en]

Challenge problems

2015-05-03T01:23:07Z

Ryan Moruzzi:

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"

! Problem 1
|- The code in this line generates a new row. You can modify the text of the row by using html code.
|One hundred mathematicians are invited to the castle of Dr. Evil for an evil math convention. Before entering the ballroom, he gives each mathematician a hat with 7 accurate digits of pi on the front. No one sees their own hat but they are all told that at least one person has 7 accurate digits of pi on their hat. Next, Dr. Evil tells them to walk into the ballroom and stand in a circle so they can see every other mathematician. He tells them the lights in the ballroom will be continuously turned off then back on and as soon as you figure out if your hat as the accurate digits of pi on the front, you must leave the room when the light is turned off.
What is the minimum number of times the lights must be turned off so that everyone will have left the room?
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"

! Problem 2
|- The code in this line generates a new row. You can modify the text of the row by using html code.
|Four cats are sitting at the four corners of a square. Each cat randomly picks a direction and starts to move along the edge of the square. What is the probability that none of the cats collide?
[[Challenge Problem 2 Solution]]
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"

! Problem 3
|- The code in this line generates a new row. You can modify the text of the row by using html code.
|You working in a nuclear power plant and have 100 switches to check. At the start of the day, each switch is turned off. You make a first pass, checking each one as you go, and if the switch is off, you flip it on; if it is on, you flip it off.

After all 100 switches have been checked, you make another pass, now only visiting every 2nd switch and proceed in the same way, i.e. if the switch is on, you flip it off and vice versa.

Again, after all 100 switches have been checked, you pass again, now visiting every 3rd switch, continuing on in the same manner as before.
This continues until you only have to visit the 100th switch.

Now, which switches will be turned on and which will be off?

|}

Challenge problems

2015-05-03T01:08:42Z

Ryan Moruzzi:

Challenge problems

2015-05-03T01:08:11Z

Ryan Moruzzi:

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"

! Problem 1
|- The code in this line generates a new row. You can modify the text of the row by using html code.
|One hundred mathematicians are invited to the castle of Dr. Evil for an evil math convention. Before entering the ballroom, he gives each mathematician a hat with 7 accurate digits of pi on the front. No one sees their own hat but they are all told that at least one person has 7 accurate digits of pi on their hat. Next, Dr. Evil tells them to walk into the ballroom and stand in a circle so they can see every other mathematician. He tells them the lights in the ballroom will be continuously turned off then back on and as soon as you figure out if your hat as the accurate digits of pi on the front, you must leave the room when the light is turned off.
What is the minimum number of times the lights must be turned off so that everyone will have left the room?
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"

! Problem 2
|- The code in this line generates a new row. You can modify the text of the row by using html code.
|Four cats are sitting at the four corners of a square. Each cat randomly picks a direction and starts to move along the edge of the square. What is the probability that none of the cats collide?
[[Challenge Problem 2 Solution]]
|}
==Problem 2 ==

Four cats are sitting at the four corners of a square. Each cat randomly picks a direction and starts to move along the edge of the square. What is the probability that none of the cats collide?

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==Problem 3 ==

Challenge problems

2015-05-03T01:06:05Z

Ryan Moruzzi:

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"

! Problem 1
|- The code in this line generates a new row. You can modify the text of the row by using html code.
|One hundred mathematicians are invited to the castle of Dr. Evil for an evil math convention. Before entering the ballroom, he gives each mathematician a hat with 7 accurate digits of pi on the front. No one sees their own hat but they are all told that at least one person has 7 accurate digits of pi on their hat. Next, Dr. Evil tells them to walk into the ballroom and stand in a circle so they can see every other mathematician. He tells them the lights in the ballroom will be continuously turned off then back on and as soon as you figure out if your hat as the accurate digits of pi on the front, you must leave the room when the light is turned off.
What is the minimum number of times the lights must be turned off so that everyone will have left the room?
|}

==Problem 1==

One hundred mathematicians are invited to the castle of Dr. Evil for an evil math convention. Before entering the ballroom, he gives each mathematician a hat with 7 accurate digits of pi on the front. No one sees their own hat but they are all told that at least one person has 7 accurate digits of pi on their hat. Next, Dr. Evil tells them to walk into the ballroom and stand in a circle so they can see every other mathematician. He tells them the lights in the ballroom will be continuously turned off then back on and as soon as you figure out if your hat as the accurate digits of pi on the front, you must leave the room when the light is turned off.

What is the minimum number of times the lights must be turned off so that everyone will have left the room?
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==Problem 2 ==

Four cats are sitting at the four corners of a square. Each cat randomly picks a direction and starts to move along the edge of the square. What is the probability that none of the cats collide?

[[Challenge Problem 2 Solution]]
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==Problem 3 ==

Challenge problems

2015-05-03T00:50:44Z

Ryan Moruzzi: /* Problem 2 */

==Problem 1==

One hundred mathematicians are invited to the castle of Dr. Evil for an evil math convention. Before entering the ballroom, he gives each mathematician a hat with 7 accurate digits of pi on the front. No one sees their own hat but they are all told that at least one person has 7 accurate digits of pi on their hat. Next, Dr. Evil tells them to walk into the ballroom and stand in a circle so they can see every other mathematician. He tells them the lights in the ballroom will be continuously turned off then back on and as soon as you figure out if your hat as the accurate digits of pi on the front, you must leave the room when the light is turned off.

What is the minimum number of times the lights must be turned off so that everyone will have left the room?
--------------------------------------------------------------------------------------------------------------------------------------------

==Problem 2 ==

Four cats are sitting at the four corners of a square. Each cat randomly picks a direction and starts to move along the edge of the square. What is the probability that none of the cats collide?

[[Challenge Problem 2 Solution]]
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==Problem 3 ==

Challenge Problem 2 Solution

2015-05-02T17:37:10Z

Ryan Moruzzi: Created page with "==Challenge Problem 2 Solution== In order for no cats to collide, all four cats must walk in the same direction. Thus, all four cats must walk clockwise OR all four cats mu..."

==Challenge Problem 2 Solution==

In order for no cats to collide, all four cats must walk in the same direction. Thus, all four cats must walk clockwise OR all four cats must walk counterclockwise. This means <math>P(</math>no cats collide<math>) = P(</math>all cats go clockwise<math>) + P(</math>all cats go counterclockwise<math>)</math>.

The probability a cat chooses clockwise or counterclockwise is <math>1/2</math>. So the probability all four cats choose clockwise or counterclockwise is <math>(1/2)*(1/2)*(1/2)*(1/2)</math>.

Therefore, <math>P(</math>no cats collide<math>) = (1/2)*(1/2)*(1/2)*(1/2) + (1/2)*(1/2)*(1/2)*(1/2) = 1/16 + 1/16 = 1/8.</math>

Challenge problems

2015-05-02T17:37:01Z

Ryan Moruzzi:

==Problem 1==

One hundred mathematicians are invited to the castle of Dr. Evil for an evil math convention. Before entering the ballroom, he gives each mathematician a hat with 7 accurate digits of pi on the front. No one sees their own hat but they are all told that at least one person has 7 accurate digits of pi on their hat. Next, Dr. Evil tells them to walk into the ballroom and stand in a circle so they can see every other mathematician. He tells them the lights in the ballroom will be continuously turned off then back on and as soon as you figure out if your hat as the accurate digits of pi on the front, you must leave the room when the light is turned off.

What is the minimum number of times the lights must be turned off so that everyone will have left the room?
--------------------------------------------------------------------------------------------------------------------------------------------

==Problem 2 ==

Four cats are sitting at the four corners of a square. Each cat randomly picks a direction and starts to move along the edge of the square.

What is the probability that none of the cats collide?
[[Challenge Problem 2 Solution]]
--------------------------------------------------------------------------------------------------------------------------------------------

==Problem 3 ==

Challenge problems

2015-05-02T17:34:07Z

Ryan Moruzzi:

==Problem 1==

One hundred mathematicians are invited to the castle of Dr. Evil for an evil math convention. Before entering the ballroom, he gives each mathematician a hat with 7 accurate digits of pi on the front. No one sees their own hat but they are all told that at least one person has 7 accurate digits of pi on their hat. Next, Dr. Evil tells them to walk into the ballroom and stand in a circle so they can see every other mathematician. He tells them the lights in the ballroom will be continuously turned off then back on and as soon as you figure out if your hat as the accurate digits of pi on the front, you must leave the room when the light is turned off.

What is the minimum number of times the lights must be turned off so that everyone will have left the room?
--------------------------------------------------------------------------------------------------------------------------------------------

==Problem 2 ==

Four cats are sitting at the four corners of a square. Each cat randomly picks a direction and starts to move along the edge of the square.

What is the probability that none of the cats collide?
[[Challenge Problem Solutions]]
--------------------------------------------------------------------------------------------------------------------------------------------

==Problem 3 ==

Answer

2015-05-02T17:32:51Z

Ryan Moruzzi: Challenge Problem Solutions

Answer

2015-05-02T17:31:10Z

Ryan Moruzzi: /* Challenge Problem 2 Solution */

==Challenge Problem 2 Solution==

In order for no cats to collide, all four cats must walk in the same direction. Thus, all four cats must walk clockwise OR all four cats must walk counterclockwise. This means <math>P(no cats collide) = P(<text>all cats go clockwise</text>) + P(all cats go counterclockwise)</math>.

The probability a cat chooses clockwise or counterclockwise is <math>1/2</math>. So the probability all four cats choose clockwise or counterclockwise is <math>(1/2)*(1/2)*(1/2)*(1/2)</math>.

Therefore, <math>P(no cats collide) = (1/2)*(1/2)*(1/2)*(1/2) + (1/2)*(1/2)*(1/2)*(1/2) = 1/16 + 1/16 = 1/8.</math>

Answer

2015-05-02T17:29:58Z

Ryan Moruzzi: Challenge Problem Answers

==Challenge Problem 2 Solution==

In order for no cats to collide, all four cats must walk in the same direction. Thus, all four cats must walk clockwise OR all four cats must walk counterclockwise. This means <math>P(no cats collide) = P(all cats go clockwise) + P(all cats go counterclockwise)</math>.

The probability a cat chooses clockwise or counterclockwise is <math>1/2</math>. So the probability all four cats choose clockwise or counterclockwise is <math>(1/2)*(1/2)*(1/2)*(1/2)</math>.

Therefore, <math>P(no cats collide) = (1/2)*(1/2)*(1/2)*(1/2) + (1/2)*(1/2)*(1/2)*(1/2) = 1/16 + 1/16 = 1/8.</math>

Challenge problems

2015-05-02T17:20:29Z

Ryan Moruzzi:

==Problem 1==

One hundred mathematicians are invited to the castle of Dr. Evil for an evil math convention. Before entering the ballroom, he gives each mathematician a hat with 7 accurate digits of pi on the front. No one sees their own hat but they are all told that at least one person has 7 accurate digits of pi on their hat. Next, Dr. Evil tells them to walk into the ballroom and stand in a circle so they can see every other mathematician. He tells them the lights in the ballroom will be continuously turned off then back on and as soon as you figure out if your hat as the accurate digits of pi on the front, you must leave the room when the light is turned off.

What is the minimum number of times the lights must be turned off so that everyone will have left the room?
--------------------------------------------------------------------------------------------------------------------------------------------

==Problem 2 ==

Four cats are sitting at the four corners of a square. Each cat randomly picks a direction and starts to move along the edge of the square.

What is the probability that none of the cats collide?
[[Answer]]
--------------------------------------------------------------------------------------------------------------------------------------------

==Problem 3 ==

Challenge problems

2015-05-02T17:14:42Z

Ryan Moruzzi: /* Problem 1 */

==Problem 1==

One hundred mathematicians are invited to the castle of Dr. Evil for an evil math convention. Before entering the ballroom, he gives each mathematician a hat with 7 accurate digits of pi on the front. No one sees their own hat but they are all told that at least one person has 7 accurate digits of pi on their hat. Next, Dr. Evil tells them to walk into the ballroom and stand in a circle so they can see every other mathematician. He tells them the lights in the ballroom will be continuously turned off then back on and as soon as you figure out if your hat as the accurate digits of pi on the front, you must leave the room when the light is turned off.

What is the minimum number of times the lights must be turned off so that everyone will have left the room?
--------------------------------------------------------------------------------------------------------------------------------------------

==Problem 2 ==

Four cats are sitting at the four corners of a square. Each cat randomly picks a direction and starts to move along the edge of the square.

What is the probability that none of the cats collide?
--------------------------------------------------------------------------------------------------------------------------------------------

==Problem 3 ==

Challenge problems

2015-04-02T20:12:07Z

Ryan Moruzzi: /* Challenge Problems */

Challenge problems

2015-04-02T20:11:48Z

Ryan Moruzzi:

=Challenge Problems=
==Problem 1==

One hundred mathematicians are invited to the castle of Dr. Evil for an evil math convention. Before entering the ballroom, he gives each mathematician a hat with 7 accurate digits of pi on the front. No one sees their own hat but they are all told that at least one person has 7 accurate digits of pi on their hat. Next, Dr. Evil tells them to walk into the ballroom and stand in a circle so they can see every other mathematician. He tells them the lights in the ballroom will be continuously turned off then back on and as soon as you figure out if your hat as the accurate digits of pi on the front, you must leave the room when the light is turned off.

What is the minimum number of times the lights must be turned off so that everyone will have left the room?