You messed with me~

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Regardless of culture, race, sex, gender or creed; humanity survives on it.

The Story of (almost) All Numbers

https://www.youtube.com/watch?v=ggOPJ8gafPo

77 posts and 55 image replies omitted. Click reply to view.The Story of (almost) All Numbers

https://www.youtube.com/watch?v=ggOPJ8gafPo

>>3730860

You forget other possibilities.

THTH…HHT

THTH…TTH

HTHT…HHT

HTHT…TTH

All of these are equally likely to occur.

You forget other possibilities.

THTH…HHT

THTH…TTH

HTHT…HHT

HTHT…TTH

All of these are equally likely to occur.

>>3730860

And, since they're just two sides of the coin arbitrarily assigned a name, the sequence:

THTHHHT is the same as HTHTTTH

It's just a matter of which side you consider to be heads or tails. Since this choice is arbitrary and doesn't depend on the coin or the throw, it must be that both outcomes TTH and HHT are equally likely. The coin cannot behave differently simply by you choosing which side is the Heads.

And, since they're just two sides of the coin arbitrarily assigned a name, the sequence:

THTHHHT is the same as HTHTTTH

It's just a matter of which side you consider to be heads or tails. Since this choice is arbitrary and doesn't depend on the coin or the throw, it must be that both outcomes TTH and HHT are equally likely. The coin cannot behave differently simply by you choosing which side is the Heads.

>>3730864

To expand to all permutations:

THTH…

HTHT…

TTTT…

HHHH…

All those have equal probability, and the next throw is independent, so the possible outcomes of the next throw are:

THTH… T

THTH… H

HTHT… T

HTHT… H

TTTT… T

TTTT… H (TTH)

HHHH… T (HHT)

HHHH… H

Both TTH and HHT are equally likely to come up on the next throw. The probability is 1/8.

To expand to all permutations:

THTH…

HTHT…

TTTT…

HHHH…

All those have equal probability, and the next throw is independent, so the possible outcomes of the next throw are:

THTH… T

THTH… H

HTHT… T

HTHT… H

TTTT… T

TTTT… H (TTH)

HHHH… T (HHT)

HHHH… H

Both TTH and HHT are equally likely to come up on the next throw. The probability is 1/8.

And again if you don't believe it, the possible outcomes, assuming you didn't get either one on the last throw are:

THTHT T

THTHT H

THTHH T (HHT)

THTHH H

HTHTT T

HTHTT H (TTH)

HTHTH T

HTHTH H

TTTTT T

TTTTT H (TTH)

HHHHH T (HHT)

HHHHH H

Two and two out of 12, which makes the probability to get HHT or TTH equal at 1/6. You can continue adding throws, and it becomes more and more likely you'll eventually get HHT or TTH, and they will always have equal probability.

THTHT T

THTHT H

THTHH T (HHT)

THTHH H

HTHTT T

HTHTT H (TTH)

HTHTH T

HTHTH H

TTTTT T

TTTTT H (TTH)

HHHHH T (HHT)

HHHHH H

Two and two out of 12, which makes the probability to get HHT or TTH equal at 1/6. You can continue adding throws, and it becomes more and more likely you'll eventually get HHT or TTH, and they will always have equal probability.

File: 1706819950327.png (189.92 KB, 800x463, 800px-HTH_v1.9_Bar.png)

>>3730847

No HTH?

No HTH?

You have 100% of one or the other occuring, assuming one or the other has occured that the next resulting occurence is either or of a 100% symmetrical difference. But also that either one variable or the other indicates itself or the other in its place. Everytime any instance of a variable occurs it denotes thousands of other results having occured in its place. That Each integer of that variable denotes an infinite possibility of arrangements for which each integer represents. Being that the chance for either to occur is infinite compound by its exampled string being infinite. With 100% data compression it actually never happened therefore its 0%.

>>3730882

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞) expanded chance (formatting infinity to match zero chance) (0% chance defined by infinite zeros of itself) Or absolute zero by absolute infinity).

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞) expanded chance (formatting infinity to match zero chance) (0% chance defined by infinite zeros of itself) Or absolute zero by absolute infinity).

50/50 chance for either H or T:

1/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

that either or variables occur

1/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

that either or variables occur

For HHT or TTH

1/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

1/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

1/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

For specifically HHT or TTH to occur between themselves.

1/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

1/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

1/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

actually i know this is wrong because both could happen so its actually.

1/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

1/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

1/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

i dont care really maybe its funny to you or something.

1/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

1/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

1/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

For specifically HHT or TTH to occur between themselves.

1/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

1/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

1/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

actually i know this is wrong because both could happen so its actually.

1/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

1/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

1/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

/

((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)/((((0/∞/∞/∞)/∞)/∞)/∞)

i dont care really maybe its funny to you or something.

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>>3730923

I feel safer already.

I feel safer already.

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>>3730847

Explanations should be as simple as possible, but no simpler.

If you get heads on the first two flips, you are guaranteed to get HHT before THH. Because the chance of starting with HH is 1/4, this is the same probability of getting HHT first. The remaining probability of 3/4 is that of running into THH first.

Explanations should be as simple as possible, but no simpler.

If you get heads on the first two flips, you are guaranteed to get HHT before THH. Because the chance of starting with HH is 1/4, this is the same probability of getting HHT first. The remaining probability of 3/4 is that of running into THH first.

>>3731050

Ultimately depends if the universe is good natured or not. this is a light-wave issue of what is observed as heads or tails. If you live in a world where light is opposed to your existence it will bely the answer, that heads is actually tails and tails is actually heads. What you get depends simply on the end resulting in favor to heads or tails, not whether or not that is true.

Ultimately depends if the universe is good natured or not. this is a light-wave issue of what is observed as heads or tails. If you live in a world where light is opposed to your existence it will bely the answer, that heads is actually tails and tails is actually heads. What you get depends simply on the end resulting in favor to heads or tails, not whether or not that is true.

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>>3731073

> If you live in a world where light is opposed to your existence

If you are living in that world then you are a vampire.

> If you live in a world where light is opposed to your existence

If you are living in that world then you are a vampire.

>>3731050

> The remaining probability of 3/4 is that of running into THH first.

No it's not. Two of the four possible throws, TH and HT may end with HHT with subsequent throws.

> The remaining probability of 3/4 is that of running into THH first.

No it's not. Two of the four possible throws, TH and HT may end with HHT with subsequent throws.

>>3731080

if h or t were pressure sensitive then appliance of pressure would indicate h or t based on sheer chance h or t occur of a change of pressure (or are they both attracted to pressure and therefore for every instance there is an h there is also a t)

if h or t were pressure sensitive then appliance of pressure would indicate h or t based on sheer chance h or t occur of a change of pressure (or are they both attracted to pressure and therefore for every instance there is an h there is also a t)

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Math Monday, nerd bitches! That means pencils down for last week's puzzle, but pick them right back up for this week's new puzzle. Congrats to >>3730613 for solving last week's puzzle. Here's the solution the author posted this morning along with your new challenge.

https://gizmodo.com/gizmodo-monday-puzzle-ugly-ducking-big-bad-wolf-1851191302

Puzzle #28: Duck Duck Go

The Ugly Duckling is in the center of a circular lake when he spots the Big Bad Wolf lurking on the shore. The wolf cannot swim, and the duckling cannot fly away from water (but can fly from land). The wolf patrols the outside of the lake waiting for the duckling to reach land so that he can feast. The wolf moves four times faster than the duckling can swim. He can see the duckling and is free to prowl along the shoreline however he wants. How can the duckling make it to land and fly away without becoming the wolf’s dinner?

As usual, this isn’t a trick question. The answer isn’t “the lake is frozen so the duckling can fly away” or anything like that. There is a legitimate strategy with mathematical reasoning behind it. Good luck!

https://gizmodo.com/gizmodo-monday-puzzle-ugly-ducking-big-bad-wolf-1851191302

Puzzle #28: Duck Duck Go

The Ugly Duckling is in the center of a circular lake when he spots the Big Bad Wolf lurking on the shore. The wolf cannot swim, and the duckling cannot fly away from water (but can fly from land). The wolf patrols the outside of the lake waiting for the duckling to reach land so that he can feast. The wolf moves four times faster than the duckling can swim. He can see the duckling and is free to prowl along the shoreline however he wants. How can the duckling make it to land and fly away without becoming the wolf’s dinner?

As usual, this isn’t a trick question. The answer isn’t “the lake is frozen so the duckling can fly away” or anything like that. There is a legitimate strategy with mathematical reasoning behind it. Good luck!

>The wolf moves four times faster than the duckling can swim.

So 500% the speed?

The longest path that the wolf has to take around the lake is pi x radius. Since the difference between pi x radius and radius is less than four times the radius - which is also how much faster the wolf is - the wolf can always catch up to the duckling. It can reach any point along the shore faster than the duckling can swim away from the center.

The "trick" only works if the wolf is less than pi times faster than the duckling, so the duck can trick the wolf to wait at the opposite side of where they're trying to swim.

So 500% the speed?

The longest path that the wolf has to take around the lake is pi x radius. Since the difference between pi x radius and radius is less than four times the radius - which is also how much faster the wolf is - the wolf can always catch up to the duckling. It can reach any point along the shore faster than the duckling can swim away from the center.

The "trick" only works if the wolf is less than pi times faster than the duckling, so the duck can trick the wolf to wait at the opposite side of where they're trying to swim.

>>3731305

Of course, the "mathematical" answer is something like aiming a certain number of degrees ahead of the wolf to maximize their running distance, but since we're dealing with a duck that can do math, we're also dealing with a wolf that knows the trick and simply switches direction, forcing the duckling to turn back the other way.

Of course, the "mathematical" answer is something like aiming a certain number of degrees ahead of the wolf to maximize their running distance, but since we're dealing with a duck that can do math, we're also dealing with a wolf that knows the trick and simply switches direction, forcing the duckling to turn back the other way.

>>3731305

wait until wolf sleeps the end. stupid question

wait until wolf sleeps the end. stupid question

>>3731305

midmaxing coast-to-shore footing differences until flight is achievable while dodging wolf counter steps. the end.

midmaxing coast-to-shore footing differences until flight is achievable while dodging wolf counter steps. the end.

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>>3731305

Saw this explained on a popular math YouTube channel, except they used different animals.

Saw this explained on a popular math YouTube channel, except they used different animals.

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https://imgur.com/gallery/4vcJ2dN

Then this just happened to their rental while they were home, they went out on the front porch from their bedroom because they heard branches cracking then this happened. My grandchildren were inside along with 2 dogs and 2 cats. Amazingly nobody was seriously hurt. Another person in a home down a ways away wasn’t as fortunate. Emotionally we’re all a mess. The tree ripped through their bedroom.

I know it sad; but what if they hired carpenters to cut parts of the fallen tree into planks and rebuild the house with it?

how much time, tools, and labor would it cost?

Then this just happened to their rental while they were home, they went out on the front porch from their bedroom because they heard branches cracking then this happened. My grandchildren were inside along with 2 dogs and 2 cats. Amazingly nobody was seriously hurt. Another person in a home down a ways away wasn’t as fortunate. Emotionally we’re all a mess. The tree ripped through their bedroom.

I know it sad; but what if they hired carpenters to cut parts of the fallen tree into planks and rebuild the house with it?

how much time, tools, and labor would it cost?

>>3731457

IF wolf stuck facing one way den you just go it the other way.

IF wolf stuck facing one way den you just go it the other way.

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Serious Math/Geometry question.

I know the symbols for perpendicular and parallel.

What is the notation for skew and intersecting lines?

Google and bing proved useless.

I know the symbols for perpendicular and parallel.

What is the notation for skew and intersecting lines?

Google and bing proved useless.

File: 1707347972413.png (92.36 KB, 3350x1886, feynman_trig_notation.png)

>>3731535

I've never seen symbols used for those cases. Should be obvious from a diagram. If you find a need for such notation, just invent your own. Worked for Feynman when he invented his own shorthand for trigonometric functions. Until he had to explain his math to others.

I've never seen symbols used for those cases. Should be obvious from a diagram. If you find a need for such notation, just invent your own. Worked for Feynman when he invented his own shorthand for trigonometric functions. Until he had to explain his math to others.

>>3731535

You know i just don't have the math on me right now.

some delta or phi/pi degree probably? really though it should be independent from the original shape and its own formula.

for me personally orbitals seem to substitute this pretty well. I have pretty much never needed to skew something and imo a true-skew it only exists as an exact mirror image to something else. for computers its like isolating an objects through a quarantine or using a filter.

I am sure you can prove it.

You know i just don't have the math on me right now.

some delta or phi/pi degree probably? really though it should be independent from the original shape and its own formula.

for me personally orbitals seem to substitute this pretty well. I have pretty much never needed to skew something and imo a true-skew it only exists as an exact mirror image to something else. for computers its like isolating an objects through a quarantine or using a filter.

I am sure you can prove it.

>>3731548

A linear or quantum skew might also be important. I think that it shows more being as an internal/external drawpoint (once you skew something the direction is important). Drawpoint for a whole formula with as its own radian, i think that describes the next step as long as they stay(or are also dependent) on a superaltive legend, you need number maps if its more than one dimension. I am just very busy.

A linear or quantum skew might also be important. I think that it shows more being as an internal/external drawpoint (once you skew something the direction is important). Drawpoint for a whole formula with as its own radian, i think that describes the next step as long as they stay(or are also dependent) on a superaltive legend, you need number maps if its more than one dimension. I am just very busy.

>>3731550

I think you can just say a vector is a skew actually. Though it helps to also define the vector zeros/limits (constraints too). Really busy.

I think you can just say a vector is a skew actually. Though it helps to also define the vector zeros/limits (constraints too). Really busy.

>>3731561

you know i think you can use say anything in (parentheses) is skewable by {bracket base skew} and [sub-base skew] then its good to pattern out over a full skew-matrix, that would work with the skew-scope. Before you do anything with it that is, you need all those parts considered to apply a vector. Then just plug it into your function(linear) and prep for quantum-skew patterns. Probably only works with uh, very basic quantum functions that have patterns. It will probably end up getting nulled anyway. Just fyi. I'm super busy and actually don't have a good example.

you know i think you can use say anything in (parentheses) is skewable by {bracket base skew} and [sub-base skew] then its good to pattern out over a full skew-matrix, that would work with the skew-scope. Before you do anything with it that is, you need all those parts considered to apply a vector. Then just plug it into your function(linear) and prep for quantum-skew patterns. Probably only works with uh, very basic quantum functions that have patterns. It will probably end up getting nulled anyway. Just fyi. I'm super busy and actually don't have a good example.

>>3731564

not sure deritives will express a skew correctly right so you need to doubledown and have normals differintiated too. There should be a tool for this. I would check wolfram and github or something.

not sure deritives will express a skew correctly right so you need to doubledown and have normals differintiated too. There should be a tool for this. I would check wolfram and github or something.

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Math Monday! Looks like no one got last week's puzzle or even made an attempt. Regardless, here's the solution and this week's new puzzles.

https://gizmodo.com/gizmodo-monday-puzzle-super-bowl-1851232384

That's right, we got a twofer:

Puzzle #29: A Sporting Chance

I missed the Super Bowl. All I know is that the two teams were perfectly matched in skill (suspend your disbelief) and that the score was not tied at halftime. I want to know the chances that the team who trailed at halftime pulled a comeback to win the game. Given only this information, what should I put the chances at?

By perfectly matched, I mean that the teams have the same probabilities of achieving various scores, and furthermore these probabilities don’t change based on game situation (i.e. which half they’re in or who’s ahead). Remember that Super Bowl games cannot end in ties: If scores are even after the second half, they enter overtime.

Bonus puzzle:

Several teams participate in a simple round-robin tournament (i.e. every team plays every other team once). Call a team a super-winner if every other team in the tournament either lost to them or lost to someone who lost to them. Argue that every tournament has at least one super-winner.

The halftime puzzle might seem like an overly simplistic model of athletic outcomes, but a study on NBA comebacks calculated how often the team that leads at halftime goes on to win the game and arrived at almost exactly the same quantity that this puzzle’s answer predicts. I’ll link the study in the solution next Monday.

https://gizmodo.com/gizmodo-monday-puzzle-super-bowl-1851232384

That's right, we got a twofer:

Puzzle #29: A Sporting Chance

I missed the Super Bowl. All I know is that the two teams were perfectly matched in skill (suspend your disbelief) and that the score was not tied at halftime. I want to know the chances that the team who trailed at halftime pulled a comeback to win the game. Given only this information, what should I put the chances at?

By perfectly matched, I mean that the teams have the same probabilities of achieving various scores, and furthermore these probabilities don’t change based on game situation (i.e. which half they’re in or who’s ahead). Remember that Super Bowl games cannot end in ties: If scores are even after the second half, they enter overtime.

Bonus puzzle:

Several teams participate in a simple round-robin tournament (i.e. every team plays every other team once). Call a team a super-winner if every other team in the tournament either lost to them or lost to someone who lost to them. Argue that every tournament has at least one super-winner.

The halftime puzzle might seem like an overly simplistic model of athletic outcomes, but a study on NBA comebacks calculated how often the team that leads at halftime goes on to win the game and arrived at almost exactly the same quantity that this puzzle’s answer predicts. I’ll link the study in the solution next Monday.

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Quadratic Reciprocity

This app provides an introduction to quadratic residues leading up to the Quadratic Reciprocity law. It also implements an algorithm for determining quadratic residues using this law. Further, when this algorithm is run it provides a step-by-step explanation of the results.

While a grasp of some advanced math is required to fully appreciate the quadratic reciprocity law, at one level it can be understood in terms of elementary number theory. Our goal is to make this exposition accessible to both levels of readers.

The app makes use of Latex and MathJax to present the mathematics with standard notation, and uses the app's programming capabilities to implement the algorithm. It does not require an internet connection and works completely offline.

https://play.google.com/store/apps/details?id=com.realandcomplexsoftware.quadraticreciprocity

This app provides an introduction to quadratic residues leading up to the Quadratic Reciprocity law. It also implements an algorithm for determining quadratic residues using this law. Further, when this algorithm is run it provides a step-by-step explanation of the results.

While a grasp of some advanced math is required to fully appreciate the quadratic reciprocity law, at one level it can be understood in terms of elementary number theory. Our goal is to make this exposition accessible to both levels of readers.

The app makes use of Latex and MathJax to present the mathematics with standard notation, and uses the app's programming capabilities to implement the algorithm. It does not require an internet connection and works completely offline.

https://play.google.com/store/apps/details?id=com.realandcomplexsoftware.quadraticreciprocity

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>>3732102

**Last Week's Puzzle #28**

Here's the video I mentioned that better explains the solution:

https://youtu.be/vF_-ob9vseM

**This Week's Puzzle #29**

**1/4 probability that the team who trailed at halftime pulled a comeback to win the game**

I reasoned by analogy using coin flips and came up with 1/4. This may have been the wrong approach, but it makes sense to me.

**Bonus Puzzle**

If there are just two teams, clearly one will be the super-winner. Given a group of three teams, there's either one team that beats the other two (*one* super-winner) or they form a cycle like in rock-paper-scissors (*three* super-winners). Add another team, and if they lose to a super-winner from the previous three matches, then the previous super-winner retains their title. If the newly added team wins against all previous super-winners, then that team becomes a super-winner. This kind of inductive reasoning shows that no matter the size of the tournament (number of teams) there will always be at least one super-winner.

Here's the video I mentioned that better explains the solution:

https://youtu.be/vF_-ob9vseM

I reasoned by analogy using coin flips and came up with 1/4. This may have been the wrong approach, but it makes sense to me.

If there are just two teams, clearly one will be the super-winner. Given a group of three teams, there's either one team that beats the other two (

File: 1708114217664.jpg (1.45 MB, 4032x3024, digraph.jpg)

>>3732394

**Correction to Bonus Puzzle Solution**

Okay, so for the case when the new team beats all the previous super-winners, the new team is not necessarily a super-winner. Here's a directed graph (digraph) to help explain. If the new team (N) loses to a team that a previous super-winner (S) lost to, then the direction of the edge connecting the bottom left node to N determines whether or not N is a potential super-winner. If N wins against this team (diagonal pointing from bottom left node to N) and this is true for all such cases, then N is a new super-winner. If N loses to this team (diagonal pointing from N to bottom left node), then S remains a super-winner. Either way, you're left with at least one super-winner.

Okay, so for the case when the new team beats all the previous super-winners, the new team is not necessarily a super-winner. Here's a directed graph (digraph) to help explain. If the new team (N) loses to a team that a previous super-winner (S) lost to, then the direction of the edge connecting the bottom left node to N determines whether or not N is a potential super-winner. If N wins against this team (diagonal pointing from bottom left node to N) and this is true for all such cases, then N is a new super-winner. If N loses to this team (diagonal pointing from N to bottom left node), then S remains a super-winner. Either way, you're left with at least one super-winner.

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Math Monday! >>3732394 solved last week's puzzle.

https://gizmodo.com/gizmodo-monday-puzzle-survivor-challenge-flags-1851232495

Puzzle #30: Survivor Flags

Jeff Probst has planted 21 flags in the ground. Tribe A and Tribe B will alternate removing either one, two, or three flags at a time (zero is not allowed). The tribe that removes the last flag wins. If Tribe A goes first, which team can force a win and what is the winning strategy?

https://gizmodo.com/gizmodo-monday-puzzle-survivor-challenge-flags-1851232495

Puzzle #30: Survivor Flags

Jeff Probst has planted 21 flags in the ground. Tribe A and Tribe B will alternate removing either one, two, or three flags at a time (zero is not allowed). The tribe that removes the last flag wins. If Tribe A goes first, which team can force a win and what is the winning strategy?

File: 1708367187068.jpg (107.4 KB, 1280x1079, fidget_the_nimbat_by_giant….jpg)

>>3732734

This is a famous game in disguise and is pretty easy to reason out, so I'll leave it for others and just mention a sneaky trick that Harry Anderson described in Games You Can't Lose:

One way to con people with this game is simple sleight of hand. You let your opponent maneuver you into a losing position, then when you go to take a 'flag' from a pile, you simultaneously drop a palmed flag into the pile, thus ruining their count. When they are sure that they are screwing you over, they are less likely to be counting accurately…

This is a famous game in disguise and is pretty easy to reason out, so I'll leave it for others and just mention a sneaky trick that Harry Anderson described in Games You Can't Lose:

One way to con people with this game is simple sleight of hand. You let your opponent maneuver you into a losing position, then when you go to take a 'flag' from a pile, you simultaneously drop a palmed flag into the pile, thus ruining their count. When they are sure that they are screwing you over, they are less likely to be counting accurately…

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>>3732734

If Tribe A goes first and utilizes the optimal strategy, they will win.

The optimal strategy is to remove only one flag on the first move. This forces Tribe B to pick from 20 remaining flags. B cannot remove enough flags so that A must pick from 16 remaining flags, but A can force this upon B on their turn. A can continue to force B into situations in which they must choose from a number of flags divisible by 4. Until B chooses from 4 remaining flags, at which point A claims victory on their subsequent move.

If Tribe A goes first and utilizes the optimal strategy, they will win.

The optimal strategy is to remove only one flag on the first move. This forces Tribe B to pick from 20 remaining flags. B cannot remove enough flags so that A must pick from 16 remaining flags, but A can force this upon B on their turn. A can continue to force B into situations in which they must choose from a number of flags divisible by 4. Until B chooses from 4 remaining flags, at which point A claims victory on their subsequent move.

File: 1708715126945.png (1.16 MB, 1187x483, prices to buy nectar.png)

Does anyone have a calculator that does 1.000 precisions so that I could accuratly calculate the price on swift nector for L$100

File: 1708715640967.png (1.34 MB, 1196x543, prices to buy nectar.png)

>>3733061

calculation follows this logic examples;

1.000 to 1.999 hits to 2.000

1 Swift Nectar is L$2.800 how much is for 100 Swift Nectars?

calculation follows this logic examples;

1.000 to 1.999 hits to 2.000

1 Swift Nectar is L$2.800 how much is for 100 Swift Nectars?

>>3733063

My standard calculator says to buy swift nectar price is 2.8 and if I buy 100 the price would be L$280 does that sound correct?

My standard calculator says to buy swift nectar price is 2.8 and if I buy 100 the price would be L$280 does that sound correct?

File: 1708718142911.png (533.26 KB, 629x514, prices to buy nectar.png)

Price L$5.800 x Flash Nectar 1000 = Price L$5,800

United States Currency $18.1250 = Price L$5.800 x Flash Nectar 1000 = Price L$5,800

Original $1 currency does not change it's own value but printing copies inflation value on other nations foriegn currencies does change their value.

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