Easy math puzzles
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53 members have voted
May 8th, 2026 at 6:42:03 PM
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Quote: Wizard
p.s. Why am I advised to capitalize Googleplex?
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I think "googolplex" is the number, and "Googleplex" is Alphabet City.
May the cards fall in your favor.
May 8th, 2026 at 9:19:01 PM
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I think that the answer depends on how the location of the portal is chosen. I assume that either path is equally likely but the probability distribution of the distance along the correct path is important (and without a rigorous definition of this we can't calculate the expected distance traveled, so we can't minimize it).
The probability distribution can't be uniform because there can't be a uniform probability distribution over an unbounded set (you can have one over an infinite set, like all the real numbers between 0 and 1, but it needs to be bounded -- there's no uniform probability distribution over all the positive reals)
The probability distribution can't be uniform because there can't be a uniform probability distribution over an unbounded set (you can have one over an infinite set, like all the real numbers between 0 and 1, but it needs to be bounded -- there's no uniform probability distribution over all the positive reals)
May 8th, 2026 at 10:46:55 PM
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This problem has already been posted twice before and never answered. Even the “googleplex” term was used. Formerly it was framed as an evil wizard in an infinite “but not completely infinite” desert. Don’t waste your time on itQuote: WizardAn Evil Wizard transports you to a distant planet, where you are placed at the beginning of two different paths. As in the Led Zeppelin song, there are two paths you may go down and there's always time to go back and switch to the other path any time you wish as many times as you wish.
Somewhere, on one of the paths, is a portal taking you back home.
The path with the portal has a finite but unknown distance (for example, an inch or a Googleplex light years). The other is infinitely long.
What strategy should you take to minimize the expected walking time to find the portal?
p.s. Why am I advised to capitalize Googleplex?
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It’s all about making that GTA
May 9th, 2026 at 12:49:32 AM
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It sounds like an interesting problem however I'm quite busy this weekend. So these are my initial thoughts.
I had assumed for simplicitiy that there was an East-West path and with one road going West and the other road going East. There were mile markers and for simplicity the prize is at one of those markers. I had assumed that initially the prize had equal probability of being East or West of the starting position (MM0).
It became apparent that the probabilities of where the prize is couldn't be uniform but must be something like the setter picks a path at random and then works their way along the path. At each Mile Marker they roll a multi-sided die and (say) if it's a 1 drops the prize otherwise moves on. This means that at MM0 the probability distribution along a path is similar to being at MM1.
As has been said you travel to a mile point and have to ascertain whether it's best to look at the next one or turn round. The upside is finding the prize at the next MM quicker, and the downside is if the prize is at all the other MMs, then you've lost 2 miles.
For instance if the setter tossed a coin, at the beginning the chance of [the prize] being at MM1E is 1/4 (MM2E=1/8 etc.). At the beginning the chances of being West is 1/2 and of being East beyond MM2E=1/8. So at MM1E the chances are now W=4/6, MM2E=1/6, MM2+E=1/6. The additional distance saved for MM2E is at least 4, and probably 6. But there is a 5/6 chance you add 2 miles to the journey. So for it's best to tun round at MM1E.
I was then going to look at different probabilities and spot a pattern, but it does look as if you turn round when the above calculation says the chances of (the downside) being West [x 2 miles] outwieghs the chances of (the upside) being further East (x >4 distance from staring position). Given you know the shape of the probaibility curve, you can now look at the differential as you change the distances involved.
This would then give an answer such you turn round when the chances of being further East is lower than p.
The problem I see is you don't yet know the turning point for the West leg, so can't tell the "upside" accurately. It seems obvious you go further along the West leg than when you first turned on the East leg. However you do know the chances of being East or West are identical when you get to the same distance.
I had assumed for simplicitiy that there was an East-West path and with one road going West and the other road going East. There were mile markers and for simplicity the prize is at one of those markers. I had assumed that initially the prize had equal probability of being East or West of the starting position (MM0).
It became apparent that the probabilities of where the prize is couldn't be uniform but must be something like the setter picks a path at random and then works their way along the path. At each Mile Marker they roll a multi-sided die and (say) if it's a 1 drops the prize otherwise moves on. This means that at MM0 the probability distribution along a path is similar to being at MM1.
As has been said you travel to a mile point and have to ascertain whether it's best to look at the next one or turn round. The upside is finding the prize at the next MM quicker, and the downside is if the prize is at all the other MMs, then you've lost 2 miles.
For instance if the setter tossed a coin, at the beginning the chance of [the prize] being at MM1E is 1/4 (MM2E=1/8 etc.). At the beginning the chances of being West is 1/2 and of being East beyond MM2E=1/8. So at MM1E the chances are now W=4/6, MM2E=1/6, MM2+E=1/6. The additional distance saved for MM2E is at least 4, and probably 6. But there is a 5/6 chance you add 2 miles to the journey. So for it's best to tun round at MM1E.
I was then going to look at different probabilities and spot a pattern, but it does look as if you turn round when the above calculation says the chances of (the downside) being West [x 2 miles] outwieghs the chances of (the upside) being further East (x >4 distance from staring position). Given you know the shape of the probaibility curve, you can now look at the differential as you change the distances involved.
This would then give an answer such you turn round when the chances of being further East is lower than p.
The problem I see is you don't yet know the turning point for the West leg, so can't tell the "upside" accurately. It seems obvious you go further along the West leg than when you first turned on the East leg. However you do know the chances of being East or West are identical when you get to the same distance.
May 9th, 2026 at 3:10:50 AM
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Quote: Ace2This problem has already been posted twice before and never answered. Even the “googleplex” term was used. Formerly it was framed as an evil wizard in an infinite “but not completely infinite” desert. Don’t waste your time on it
I'm fairly sure that the point I made above is the reason it hasn't been answered (the question is not well defined) but I'd like to get your feedback (particularly if I'm wrong)
May 9th, 2026 at 6:15:40 AM
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Quote: WizardQuote: ThatDonGuyLabel the two paths A and B, and let A(x) and B(x) be the points on each path that are distance x from the starting point.
Start by walking on path A, then the turnaround points are A(1), B(2), A(3), B(4), A(5), B(6), and so on.
The exact distance from, say, 0 to A(1) is irrelevant.
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You are right about establishing turnaround points. I would say the distances between them are very relevant. That is the point of the problem. For example, if you walked 100 miles down path A and turned around and eventually returned to path A, after not finding the portal on path B, then you wouldn't turn around again at 101 miles from the junction.
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If every mile of each path is essentially equally probable to contain the portal then after taking a hundred years to walk 1,000,000 miles down an infinite path why would you turn around and spend the next 100 years walking back to your point of origin? It would have profited you more to spend that next 100 year increment walking another 1,000,000 miles down the first path because you would have a chance to find the portal during that time.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
May 9th, 2026 at 9:09:41 AM
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Solution to the Evil Wizard/Two Paths Puzzle
You have been walking down the infinite path for some unremembered number of years when you find the Cemetery of Forgotten Books, an infinite labyrinth of rooms with shelves that contain every book ever written by humanity -whether or not it was ever published. You start browsing through the library and encounter a young woman - a young Sydney Sweeney. For this is an infinitely long path that you may walk for an infinite amount of time, and surely one can encounter anything on such a path?
And young Sydney Sweeney walks toward you, smiling -beaming with happiness - saying "It has been so long since I have seen another human being." Her eyes look down in momentary shyness and she continues "And I have always been attracted towards older men, particularly nerdy men of great intelligence." She then walks into the oval of your arms and presses her body against yours. She says "I've been so lonely. Please stop your endless search for the 'Portal to Home' long enough to spend a night or two with me."
She tilts her head up. She is young, beautiful. She holds the world in her eyes.
The two of you kiss, at first hesitantly then passionately. Nature takes its course.
You awake the next morning and Sydney is standing there with a tray in her hands, serving you breakfast in bed. Bread, exotic cheeses, fresh fruit, eggs, bacon. And the smell of fresh apple pie coming from the next room. Sydney smiles, "Yes. I also enjoy baking."
Ten years have passed. You sit in a seat close to the fire, with Sydney by your side, reading an unpublished work by JRR Tolkien. Near you, waiting to be read, are unpublished works by Dostoyevski and Lee Child. But suddenly, a road-weary Mike Shackleford walks into the room and looks at you quizzically. "Why are you just sitting there?" he asks. "You should be looking for the Portal to Home."
"But I am home." you reply.
"My puzzle statement says that you are looking for a Portal to Home."
You say. "The puzzle statement also says that we live forever and have been given the gift of an infinitely long path to walk. It is folly to look for a portal. I believe we should keep walking until we find our heart's desire and then stop for as long as we want."
Michael says "But that's not the statement of the puzzle."
Sydney speaks up. " Well, I have a lonely sister who lives here who would be perfect for you. She is better looking than me, intelligent, a great conversationalist, she has a great wine cellar and a two word tattoo on the small of her back, right above her butt cheeks.
"A two word tattoo? I do love word puzzles, so very much. Tell me, what does her tattoo say?"
And Sydney replies :
Michael says " Uh...ahem. Very tempting. But I am a mathematician and I need to be true to my puzzle."
Sydney counters with "Well, our Cemetery of Forgotten Books has lost letters about mathematics by Euler, translated into English. There's an unpublished book with a simple proof of the Riemann Hypothesis. Lost books on combinatorial theory."
Shaken, Michael says: "But there's a portal to home somewhere on these two paths. And you know I like to hike."
"Yep" she says "and just a short hike down the road there's a Cemetery of Forgotten License Plates. Infinite labyrinth thingy, goes on forever."
The room grows silent and the silence stretches for minutes.
At that moment Sydney's sister walks into the room, and she can indeed make blind men see. She turns to the Wizard and says " Sir, will you be my hero?"
And the Wizard, ever the gentleman, says
So you see, dear reader, some puzzle statements are so fantastic that men can find wisdom in their solutions in unexpected ways.
You have been walking down the infinite path for some unremembered number of years when you find the Cemetery of Forgotten Books, an infinite labyrinth of rooms with shelves that contain every book ever written by humanity -whether or not it was ever published. You start browsing through the library and encounter a young woman - a young Sydney Sweeney. For this is an infinitely long path that you may walk for an infinite amount of time, and surely one can encounter anything on such a path?
And young Sydney Sweeney walks toward you, smiling -beaming with happiness - saying "It has been so long since I have seen another human being." Her eyes look down in momentary shyness and she continues "And I have always been attracted towards older men, particularly nerdy men of great intelligence." She then walks into the oval of your arms and presses her body against yours. She says "I've been so lonely. Please stop your endless search for the 'Portal to Home' long enough to spend a night or two with me."
She tilts her head up. She is young, beautiful. She holds the world in her eyes.
The two of you kiss, at first hesitantly then passionately. Nature takes its course.
You awake the next morning and Sydney is standing there with a tray in her hands, serving you breakfast in bed. Bread, exotic cheeses, fresh fruit, eggs, bacon. And the smell of fresh apple pie coming from the next room. Sydney smiles, "Yes. I also enjoy baking."
Ten years have passed. You sit in a seat close to the fire, with Sydney by your side, reading an unpublished work by JRR Tolkien. Near you, waiting to be read, are unpublished works by Dostoyevski and Lee Child. But suddenly, a road-weary Mike Shackleford walks into the room and looks at you quizzically. "Why are you just sitting there?" he asks. "You should be looking for the Portal to Home."
"But I am home." you reply.
"My puzzle statement says that you are looking for a Portal to Home."
You say. "The puzzle statement also says that we live forever and have been given the gift of an infinitely long path to walk. It is folly to look for a portal. I believe we should keep walking until we find our heart's desire and then stop for as long as we want."
Michael says "But that's not the statement of the puzzle."
Sydney speaks up. " Well, I have a lonely sister who lives here who would be perfect for you. She is better looking than me, intelligent, a great conversationalist, she has a great wine cellar and a two word tattoo on the small of her back, right above her butt cheeks.
"A two word tattoo? I do love word puzzles, so very much. Tell me, what does her tattoo say?"
And Sydney replies :
Breed Me.
Michael says " Uh...ahem. Very tempting. But I am a mathematician and I need to be true to my puzzle."
Sydney counters with "Well, our Cemetery of Forgotten Books has lost letters about mathematics by Euler, translated into English. There's an unpublished book with a simple proof of the Riemann Hypothesis. Lost books on combinatorial theory."
Shaken, Michael says: "But there's a portal to home somewhere on these two paths. And you know I like to hike."
"Yep" she says "and just a short hike down the road there's a Cemetery of Forgotten License Plates. Infinite labyrinth thingy, goes on forever."
The room grows silent and the silence stretches for minutes.
At that moment Sydney's sister walks into the room, and she can indeed make blind men see. She turns to the Wizard and says " Sir, will you be my hero?"
And the Wizard, ever the gentleman, says
Ma'am, you had me at Breed Me
So you see, dear reader, some puzzle statements are so fantastic that men can find wisdom in their solutions in unexpected ways.
Last edited by: gordonm888 on May 9, 2026
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
May 9th, 2026 at 12:08:55 PM
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Quote: WizardQuote: ThatDonGuyLabel the two paths A and B, and let A(x) and B(x) be the points on each path that are distance x from the starting point.
Start by walking on path A, then the turnaround points are A(1), B(2), A(3), B(4), A(5), B(6), and so on.
The exact distance from, say, 0 to A(1) is irrelevant.
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You are right about establishing turnaround points. I would say the distances between them are very relevant. That is the point of the problem. For example, if you walked 100 miles down path A and turned around and eventually returned to path A, after not finding the portal on path B, then you wouldn't turn around again at 101 miles from the junction.
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It seems a unit vector is needed, if this is going to be a problem that includes proximation and quantization of distances. For example, you mentioned coming out of a subway and looking for your hotel knowing it was on that street but could be in either direction. The first thing I would do is look in both directions hoping I could see it, using sight as an alternative to motion. How far you can see, thus how close you have to be to the portal to have "found it" seem like they should have some effect on the turnaround points.
