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Medical Forum / Diseases and Disorders / Cancer / April 2008puzzle for math nerds
Next time you're sitting there, plugged in, waiting for the poisons to drip in, think about this one: How can you completely subdivide a square into smaller squares, such that no squares overlap, no space in the larger square is left out of the smaller squares, and the total number of smaller squares is a prime number? Note that the smaller squares don't have to all be the same size---the problem would be trivial if they did. As an example, you could place two half-size squares with three third-size squares on top of them, making for five (prime!) total. Unfortunately, while this would cover the whole width of the larger square, it would only cover 5/6th the height, so this example fails. OK, maybe that counts as a gimme, but what other examples work? Stuff like this (and Sudoku, crosswords, reading short stories) helped me get through the long hours I couldn't sleep through treatments. I hope it helps someone else. FWIW, I can now (15 mo. after last treatment) think about the whole mess, even that noise the pump made, without the sense of disgust it used to bring. Time wounds all heels. --- CSM | Next time you're sitting there, plugged in, waiting for the poisons to | drip in, think about this one: [quoted text clipped - 18 lines] | --- | CSM I think I've got it, CSM. It's one of those puzzles that make you kick yourself because the answer is in the example. Quite ingenious! I'll say no more. It's very good to hear of your recovery and the sturdy way you now look back on the dark times. Are you doing anything the prevent a recurrence? Do you believe you know what caused it in the first place? "Once the crab gets you in its pincers it will not let go". [Cancer Ward:A Solzhenytsin]. Do you believe that? I don't,but I do believe there must be research and reform on our own part to retain our good health. Thanks to the miracle of modern surgery you are well once more,but if it was your old lifestyle that brought it on in the first place,then there will need to be changes made. I hope you will accept these comments in the spirit in which they are offered -- to help. With Every Good Wish, Old Bill. > <csm7...@hotmail.com> wrote in message > [quoted text clipped - 25 lines] > yourself > because the answer is in the example. Quite ingenious! I'll say no more. I'm putting a bit about that at the bottom of the message, where those not wishing to spoil the puzzle can easily skip it. > It's very good to hear of your recovery and the sturdy way you now look back > on the dark times. > > Are you doing anything the prevent a recurrence? Thanks. I'm on a schedule for colonoscopies which should catch any recurrence before it becomes cancerous. I'm also on a schedule for PET/CT scans to catch mets, hopefully early. > Do you believe you know what caused it in the first place? Colon cancer runs in my family. I may have increased my risk with a diet rich in red meats and other evil foods. I can't say I've mended my wicked ways, though. Since I didn't have any polyps but the one that went nonlinear, my future risk is probably minimal. > "Once the crab gets you in its pincers it will not let go". > [Cancer Ward:A Solzhenytsin]. [quoted text clipped - 8 lines] > With Every Good Wish, > Old Bill. I don't believe Solzhenytsin (what an angry man!), but it was much more true when he wrote it. I get more fiber than I used to, which may help reduce my risks, but I still enjoy barbecue and seared steak. According to statistics, I have about a 50% chance of eventual death, but it will likely not be from cancer. And now for the puzzle part: SPOILER ALERT There are multiple models that solve the problem, but I thought the example made for an amusing one. There's likely a unifying model for all solutions, but I've spent too long away from real math to see it. For another model, consider the case of a square size x made of unit squares, then replacing subsets of those with larger squares. If the larger squares are of size y, each replacement reduces the total number of squares by SQR(y) - 1. There are a lot of cases where SQR(x) - (n * (SQR(y) - 1)) is prime. One example is x=5, y=2, n=2. This gives a square of length 5 subdivided into 2 (2 x 2) squares and 17 (1 x 1) squares, for a total of 19 squares. For someone like me, working through the examples, and trying to generalize them, makes time in the chair pass more swiftly. As long as the fog doesn't make it all drift out of reach, that is. | > <csm7...@hotmail.com> wrote in message news:9fad764c-4d29-4b7d-b7b1-1807fbdf569c@b64g2000hsa.googlegroups.com... | > | Next time you're sitting there, plugged in, waiting for the poisons to | > | drip in, think about this one: [snip] | > | --- | > | CSM [snip] | > It's very good to hear of your recovery and the sturdy way you now look back | > on the dark times. [quoted text clipped - 11 lines] | my wicked ways, though. Since I didn't have any polyps but the one | that went nonlinear, my future risk is probably minimal. I do hope it is,CSM but I have heard the expression "runs in the family" used in many contexts each intended to inply genetic heredity and hence "inescapable destiny". For example,I know an obese family,mother and all the children,all obese."Oh,it runs in the family", is the explanation,meaning it's in the genes, whereas it's congenital,the kids are brought up with the same eating habits as the mother. Change the diet,eliminate the obesity. | > "Once the crab gets you in its pincers it will not let go". | > [Cancer Ward:A Solzhenytsin]. | > | > Do you believe that? I don't,but I do believe there must be research | > and reform on our own part to retain our good health. [snip] | > With Every Good Wish, | > Old Bill. [quoted text clipped - 4 lines] | steak. According to statistics, I have about a 50% chance of eventual | death, but it will likely not be from cancer. I think you meant to write something else didn't you? The rest of us have a 100% of eventual death :-) | (A 50% chance of total remission,perhaps? Careful with your answer CSM;if it's yes I shall ask "what about the other 50%") | And now for the puzzle part: SPOILER ALERT | | [snip] I'm afraid you have left me at the starting gate with your maths. My main mental exercise is a daily "Countdown" on Channel 4. A bit jollier too,but I understand your intention, and the fortitude you have shown unquestionably deserves the best outcome. Good Luck in the saddle, and in Life. Old Bill. > <csm7...@hotmail.com> wrote in message > [quoted text clipped - 61 lines] > Careful with your answer CSM;if it's yes > I shall ask "what about the other 50%") That was a sort of inside joke. Something close to half the people who have ever been born are alive today (I may have the number a bit off, but that's the whole point of stats---making things up as you go along). Interpreted one way, that would mean that we who are alive today have a 50% chance of immortality. In fact, if you analyze the data further, you'll find that people born in the eighteenth century or before are all dead, those born in the nineteenth are nearly all dead, and that a progressively larger proportion of people born in later decades are still alive. So it appears that those born in the latter half of the twentieth century, as I was, have a low risk of death. All this, and the insurance companies won't write new policies for me. Go figure. > | And now for the puzzle part: SPOILER ALERT > | [quoted text clipped - 9 lines] > > Old Bill. Math nerd puzzles, like our jokes, often leave most people flat. That's OK. As long as somebody got something useful or diverting from it, it was worth writing. Someone with that kind of mind can spend a good while on this thinking about the nature of primes, squares, etc. --- CSM | > <csm7...@hotmail.com> wrote in message news:7d7e35ab-1038-4abe-9df8-427f378cfe77@a23g2000hsc.googlegroups.com... | > | On Apr 3, 2:29 pm, "Old Bill" <matb...@yahoo.com> wrote:| > <csm7...@hotmail.com> wrote in message | > >news:9fad764c-4d29-4b7d-b7b1-1807fbdf569c@b64g2000hsa.googlegroups.com... | > | > | Next time you're sitting there, plugged in, waiting for the poisons to | > | > | drip in, think about this one: [SNIP] | > According to statistics, I have about a 50% chance of eventual | > | death, but it will likely not be from cancer. [quoted text clipped - 17 lines] | death. All this, and the insurance companies won't write new policies | for me. Go figure. Got it,CSM.I rose to the bait. So when it gets to 0% our progeny become ill-conceived,presumably. | > | And now for the puzzle part: SPOILER ALERT | > | [quoted text clipped - 10 lines] | --- | CSM You must watch your diet,CSM. Seriously. You clearly know yours is not right. Life is still liveable without prime beefsteak. I'll say no more. Old Bill. squares only, or do rectangles count? [yeah - i'm guessing that would be way too easy?] > Next time you're sitting there, plugged in, waiting for the poisons to > drip in, think about this one: [quoted text clipped - 18 lines] > --- > CSM On Apr 3, 2:49 pm, "Paul T. Holland" <pholl...@bellatlantic.net> wrote: > squares only, or do rectangles count? [yeah - i'm guessing that would be > way too easy?] I think so. Allowing either the smaller units or the overall shape to be non-square makes the problem pretty much go away. If you put any prime number of equal squares next to each other, you get a rectangle, and you can divide any square into any prime number of rectangles. |
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