tag:blogger.com,1999:blog-5730391.post4371574181094862556..comments2024-04-12T03:59:51.417-07:00Comments on Blaine's Puzzle Blog: NPR Sunday Puzzle (Jun 29): Anyone have a Pencil?Blainehttp://www.blogger.com/profile/06379274325110866036noreply@blogger.comBlogger63125tag:blogger.com,1999:blog-5730391.post-71504872041685568012008-08-01T03:45:00.000-07:002008-08-01T03:45:00.000-07:00My only point in posting the pets puzzle wasn't to...My only point in posting the pets puzzle wasn't to see if anyone could solve it. Obviously it was nearly identical - nearly so - so if you could do pencils, then pets was a walk in the park - which is what I meant in my quip, "Pencils or pets, it's all about the same (almost)." (Almost) referring to slightly different coefficients.<BR/><BR/>However, when I saw it in Click and Clack's archives (as I tried to cheat on 335 443 554), I realized that I may have just encountered it before. (Sometimes I never miss CarTalk and doing their puzzler, other times I forget they and them even exist). But I never would have forgotten Linear Diophatine Equations (which, to the best of my aging memory, never even came up in HS or 4 years excelling at RPI).<BR/><BR/>So, I just wanted to highlight the fact that these sorts of puzzlers come in many flavors and come up every so many years, here and there, in one form or another, as I had learned from reading about this very interesting class of problems (Thanks to Blaine's Link and then Wiki - knowing what they are called made all the difference). <BR/><BR/>Of course, t'were I more passionate about doing puzzlers and solving everything, everywhere, all the time (like my physicist friends are and do), than simply talking about puzzlers, I would have solved the Pets one too. <BR/><BR/>But, to me, the real fun is the self discovering how, not the simply doing, when you already know how. At that point, it's just work, and work I like to be paid for (even if it is fun too). }:-)gregdavidhttps://www.blogger.com/profile/13511045845739028402noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-64159778526904055572008-07-30T15:45:00.000-07:002008-07-30T15:45:00.000-07:00Unlike our puzzle, no fractional purchases require...Unlike our puzzle, no fractional purchases required.Benhttps://www.blogger.com/profile/04754156636762779545noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-53713351299464741462008-07-30T15:35:00.000-07:002008-07-30T15:35:00.000-07:00You can solve it the exact same way as the pencils...You can solve it the exact same way as the pencils...<BR/><BR/>Let D, C and M be the number of dogs, cats and mice, respectively.<BR/><BR/>Number:<BR/>D + C + M = 100<BR/><BR/>Cost:<BR/>15D + C + M/4 = 100<BR/><BR/>Multiply the second equation by 4 to get rid of fractions:<BR/>60D + 4C + M = 400<BR/><BR/>Subtract the first equation to eliminate M:<BR/>59D + 3C = 300<BR/><BR/>Valid values of D are 1 to 5. But only one of these gives a value of C which is a whole number. The rest is left as an exercise for the reader.Blainehttps://www.blogger.com/profile/06379274325110866036noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-22970219873553769962008-07-30T15:06:00.000-07:002008-07-30T15:06:00.000-07:00Pencils or pets, it's all about the same (almost)....Pencils or pets, it's all about the same (almost). I just found this in the CarTalk Puzzler Archive:<BR/><BR/>New Puzzler: Cats, Dogs and Mice - Oh My!<BR/><BR/>RAY: I happened to find this puzzler the other day in my puzzler folder. It's from Barry Lawber. And it's from July '96. <BR/><BR/>TOM: I hope he's still alive.<BR/><BR/>RAY: I hope so too. <BR/><BR/>Here's his puzzler:<BR/><BR/>You're given a hundred dollars and told to spend it all purchasing exactly a hundred animals at the pet store. Dogs cost $15. Cats cost a buck, and mice are 25 cents each. <BR/><BR/>TOM: Let me get this straight. You have to spend exactly a hundred bucks and you end up with exactly a hundred animals?<BR/><BR/>RAY: Right. The other only other criterion is that you have to purchase at least one of each animal.<BR/><BR/>The question is, how many of each animal do you have to purchase to equal a hundred animals purchased at exactly a hundred dollars?gregdavidhttps://www.blogger.com/profile/13511045845739028402noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-72047506143753704742008-07-07T10:20:00.000-07:002008-07-07T10:20:00.000-07:00I was a bit disheartened that Will simply gave the...I was a bit disheartened that Will simply gave the answer to this puzzler and did not mention one iota about the solution, the pitflls and/or that such problems exist in a special realm of mathematics that have puzzled and challenged even the greatest of mathematicians for centuries, even to this very day.<BR/><BR/>From Wiki:<BR/><BR/>"The questions asked in Diophantine analysis include:<BR/><BR/>Are there any solutions? <BR/>Are there any solutions beyond some that are easily found by inspection? <BR/>Are there finitely or infinitely many solutions? <BR/>Can all solutions be found, in theory? <BR/>Can one in practice compute a full list of solutions? <BR/>These traditional problems often lay unsolved for centuries, and mathematicians gradually came to understand their depth (in some cases), rather than treat them as puzzles."<BR/><BR/>Clearly Will is a true puzzle guy first and foremost, and not a mthematician, and thus, has failed to understand the greater depth of what he does. Nobody's perfect.gregdavidhttps://www.blogger.com/profile/13511045845739028402noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-45088347366567220292008-07-04T01:34:00.000-07:002008-07-04T01:34:00.000-07:00What a dumb puzzle. Showme a 1/4-cent piece,1/2-c...What a dumb puzzle. Show<BR/>me a 1/4-cent piece,1/2-cemt piece?gerihttps://www.blogger.com/profile/08699362527317413011noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-32121658343982991562008-07-03T14:17:00.000-07:002008-07-03T14:17:00.000-07:00This comment has been removed by the author.Natashahttps://www.blogger.com/profile/14139505187498448200noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-9756340776299923042008-07-03T13:16:00.000-07:002008-07-03T13:16:00.000-07:00You should have made that your own puzzler. Somet...You should have made that your own puzzler. Something like: What does the Pencil Puzzler and one or more of those word sets have in common? That would have been a real good one. I doubt I would have make the leap to alphabetic positions. <BR/><BR/>It would be neat to formulate a similar puzzle that has items that start with those letters (instead of pencils) and see if anyone catches it. Like a mystery novel, all the clues are there in plain view, you just have to know where to look (viz., shift your focus).<BR/><BR/>Enjoy the BBQ. Probably Marinated Pork Steaks, COB and marinated and grilled veggies and gourmet potatos here, plus some sort of spirits (maybe). No clues there, just love to grill in the Spring, Summer and Fall and just anything can be done on one, and it keeps the house cooler to boot.gregdavidhttps://www.blogger.com/profile/13511045845739028402noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-14595792657088311152008-07-03T12:39:00.000-07:002008-07-03T12:39:00.000-07:00Anyone notice my clues to the answer?Conan O'Brien...Anyone notice my clues to the answer?<BR/><BR/>Conan O'Brien (COB)<BR/>Cheese Omelette + Bacon (COB)<BR/>Chicken Or Burgers (COB)<BR/>Corn on the COB<BR/><BR/>Taking the position of each letter in the alphabet:<BR/>C = 3<BR/>O = 15<BR/>B = 2Blainehttps://www.blogger.com/profile/06379274325110866036noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-69591514287158983622008-07-03T10:49:00.000-07:002008-07-03T10:49:00.000-07:00Everyone ready for 4th of July? I'm trying to fig...Everyone ready for 4th of July? I'm trying to figure out if I should barbecue chicken or burgers. Corn on the cob is a given, though.Blainehttps://www.blogger.com/profile/06379274325110866036noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-84465520032879043562008-07-03T08:34:00.000-07:002008-07-03T08:34:00.000-07:00In reply to:"i just wish the quantity of cents wer...In reply to:<BR/><BR/>"i just wish the quantity of cents were all whole numbers :("<BR/><BR/>It can be (sort of). Just multiply your total cost equation by 4 (on both sides). All the nasty fractions (or .25 and .5) go away and, luckily with this puzzle at least, no ugly fractions return. If you do it right, then you will find doing a spreadsheet cost your far more time than it saved.<BR/><BR/>BTW, I really wish Will Shortz had mentioned the whole pencil requirement but it really isn't much of a stretch as far as assumptions go. It's the belief that you don't have to buy and 2's and 4's of the cheap ones that most have problem coming to terms with (which is surely an interesting phenomenon to study in the realm of Psychology, Reason and Marketing).gregdavidhttps://www.blogger.com/profile/13511045845739028402noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-76019067182289263762008-07-03T07:44:00.000-07:002008-07-03T07:44:00.000-07:00occasionally i search on Google for forums or bull...occasionally i search on Google for forums or bulletin boards where people are discussing the Sunday Puzzle, and i'm so glad to find one such as this!<BR/><BR/>two coworkers and i hammered away at the pencils puzzle, and ultimately we agree with GregDavid, that the focus should be limited to the pencils as integers, the ability to buy any number of pencils, despite how they're priced. <BR/><BR/>i just wish the quantity of cents were all whole numbers :(<BR/>it would have made it more satisfying to solve the riddle, especially after making an Excel spreadsheet with formulas in each cell, etc. oh well.minncognitohttps://www.blogger.com/profile/14800835601771464544noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-46240177081997070082008-07-03T04:40:00.000-07:002008-07-03T04:40:00.000-07:00Three things:1. Blaine's math link is good. I'm ...Three things:<BR/><BR/>1. Blaine's math link is good. I'm only on page 2 and I can already see I will enjoy reading it. Isn't it funny how as adults we sometimes wish we could go back to school and just enjoy learning where it was so often a chore at the time?<BR/><BR/>2. I post the NPR puzzle on my blog every week and offer hints to whoever wants to email me for one. Sometimes (Shalit, Ebert) the emails are not particularly interesting, but with certain puzzles such as 8735 and this week's, it can lead to some unexpected and surprising revelations.<BR/><BR/>For example, one correspondent came up with a solution in which the man bought fractional pencils:<BR/><BR/>"hey, i think i got it....<BR/> <BR/> He bought 3.5 of A<BR/> 7.5 of B<BR/> and 9 of C<BR/> <BR/>A + B + C = 20<BR/>3.5 + 7.5 + 9 = 20<BR/>4A + 1/2B + 1/4C = 20<BR/>14 + 3.75 + 2.25 = 20<BR/> <BR/>its right =D but i dont get how you can buy 3.5 of a pencil =\ is there a different way?"<BR/><BR/>I told him I believed the intended answer involved round numbers of pencils, if not costs, but who knows, maybe they'd accept his answer too.<BR/><BR/>3. I have not tried to solve the bus puzzle yet. At first glance it looked to me like a pretty straightforward exercise in determining at what physical point the express bus' faster rate compensated for its later start. But Greg David's comment just above this one makes me wonder whether it is tougher than I first thought?Benhttps://www.blogger.com/profile/04754156636762779545noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-22331912282669912092008-07-03T03:42:00.000-07:002008-07-03T03:42:00.000-07:00Thanks for the excellent math link Blaine. That i...Thanks for the excellent math link Blaine. That is highly educational. Sorry for being too lazy to look it up myself.**<BR/><BR/>** Then again, I never would have pulled DIOPHANTINE out of my proverbial rear so not sure what I would have searched on. But clearly the key is simultaneous equations with integral solutions. I kind of intuited that the whole pencil condition was key and is surely what made a singular solution even possible but I still seemed to believe that condition could be used to eliminate trial and error rather than just minimize it. Otherwise why would I have multiplied through by 4 to get rid of the halves and quarters(a stroke of real genius that I didn't even realize at the time).<BR/><BR/>Still struggling with the Bus Problem. I think I have the answer, through logic, intuition and some graphical aids, but I have yet to try to prove it mathematically (a bit out of laziness but also from a lack of a definitive method and the requsite genius to come up with one).gregdavidhttps://www.blogger.com/profile/13511045845739028402noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-36941982169130478702008-07-03T00:36:00.000-07:002008-07-03T00:36:00.000-07:00It is a little sad to be up in the middle of the n...It is a little sad to be up in the middle of the night working on a puzzler, but I think I got it. I set up a grid of 5x4 squares, each square representing a pencil. I used trial and error to assign each square a price until they added up to 20 cents. Using a visual made it easier for me.Linhttps://www.blogger.com/profile/11907714232479896561noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-58787671682775643642008-07-02T23:42:00.000-07:002008-07-02T23:42:00.000-07:00Yet another comment . . . ;-)This being a puzzle f...Yet another comment . . . ;-)<BR/>This being a puzzle from a 19th century trade card, I wondered what coins were available. The half cent coin was produced in the United States from 1793-1857.<BR/>http://en.wikipedia.org/wiki/United_States_half_cent_coinJoe Whttps://www.blogger.com/profile/18115545643638113334noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-23818945571819154262008-07-02T17:36:00.000-07:002008-07-02T17:36:00.000-07:00This problem, with two equations and three unknown...This problem, with two equations and three unknowns (where the unknowns are integers) fails into a category called <A HREF="http://www.geometer.org/mathcircles/diophantine.pdf" REL="nofollow">Linear Diophantine Equations</A>. In our case, the easiest solution is reduce the problem to one equation with two unknowns and then try all the possibilities of one variable (eg. A, the number of 4 cent pencils).Blainehttps://www.blogger.com/profile/06379274325110866036noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-53702378313840429112008-07-02T15:13:00.000-07:002008-07-02T15:13:00.000-07:00Here is the matrix formula:http://ccrma.stanford.e...Here is the matrix formula:<BR/><BR/>http://ccrma.stanford.edu/~jos/mdft/Solving_Linear_Equations_Using.html<BR/><BR/>By subbing for c, you get 2 linear equations, and you can solve from there.Don Hodunhttps://www.blogger.com/profile/10629317119387671011noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-25635394339775391522008-07-02T11:32:00.000-07:002008-07-02T11:32:00.000-07:00Sorry, I'm the one that brought up the Omelet Puzz...Sorry, I'm the one that brought up the Omelet Puzzler. It seemed fittig to mention it because not only did Will give us a rare math puzzler, but so did CarTalk, and Will's was a real puzzler (challenging) and Click and Clack's was hardly a puzzler at all (which they admitted at the outset).<BR/><BR/>Spoiler alert: I'm all about getting off topic. Sorry!gregdavidhttps://www.blogger.com/profile/13511045845739028402noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-6200463664910317382008-07-02T11:21:00.000-07:002008-07-02T11:21:00.000-07:00Sorry, I guess my comments about Conan O'Brien and...Sorry, I guess my comments about Conan O'Brien and the cheese omelette w/ bacon (ala Car Talk Puzzler) were a little off-topic. I'll try to stick to comments relevant to the pencil puzzle.Blainehttps://www.blogger.com/profile/06379274325110866036noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-76937566235971095502008-07-02T11:18:00.000-07:002008-07-02T11:18:00.000-07:00It's funny how our brains work. We see 4 for a do...It's funny how our brains work. We see 4 for a dollar and we isntanly assume we have to buy them 4 at a time. <BR/><BR/>Marketers know this all too well and use it to their advantage to sell more product. That is why they so often have two-for deals. They know that most think they have to buy two when they really don't. <BR/><BR/>I, of course, nearly always just buy one which will then ring up at half the 2-for price. Great way to try and get the store's computer system to mess up and then get the item for free as a courtesy for scanner errors. Plus, before there were automatic UPC scanners, the checkout person would have to do division and you know how few of them can actually do any math anymore.gregdavidhttps://www.blogger.com/profile/13511045845739028402noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-69297050826376080632008-07-02T11:16:00.000-07:002008-07-02T11:16:00.000-07:00I like the elegance of one of the middle moves Gre...I like the elegance of one of the middle moves Greg David made in his math approach, i.e. substituting in that expression from one equation into the other one. It led more directly to the answer. <BR/><BR/>Failing to see that, I still had viable equations but it took more trial and error work to reach the answer.Benhttps://www.blogger.com/profile/04754156636762779545noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-75139976090616913672008-07-02T11:13:00.000-07:002008-07-02T11:13:00.000-07:00Carson famously had pencils with erasers at both e...Carson famously had pencils with erasers at both ends, which he would idly drum with or toss and catch. But I think Letterman more than Carson was known as a thrower. For one thing Letterman has the fake windows and they play the crashing glass sound when he throws... wait a minute... maybe I was thinking of Letterman throwing his blue index cards through the window.Benhttps://www.blogger.com/profile/04754156636762779545noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-90150301475875416162008-07-02T11:08:00.000-07:002008-07-02T11:08:00.000-07:00My first post here. It's interesting reading all ...My first post here. It's interesting reading all the clues and thoughts. For some reason, my head found it much easier to think of these as three distinct items (like apples, bananas and peaches) and not all pencils. <BR/>And, yes it is easy if you allow that you can buy the pencils in "odd lots" -- after all, we can buy a gallon of gas for $4.0397. <BR/>Late night TV is a distant memory for me, but I would have sworn it was Johnny Carson who tossed the pencil over his shoulder every night.JonBhttps://www.blogger.com/profile/13861302417964277065noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-37806987505034144402008-07-02T11:07:00.000-07:002008-07-02T11:07:00.000-07:00Sorry about posting the spoiler. I was feeling a b...Sorry about posting the spoiler. I was feeling a bit proud of my latest uber-streamlined method** (I solved this long before I even found your blog, the hard way.)<BR/><BR/>** (My friend and I have been bantering back and forth about the best way to solve this (without Excel). He sent me something that seemed novel but quite convoluted. That sparked something in me to work out a better solution, somewhat based on his, that was very staightforward and quick easy to work out. Instead of my original brute force trial and error method that required many cases to be calculated. Worked? yes! Quick and painless, no!)<BR/><BR/>I'm still wondering if anyone knows a way (or the way) to solve this without having to do any trial and error or iterations, you know, purely with equations and no test substitutions? Or is it that problems like this really do exist beyond the realm of "pure math?" Just wondering.gregdavidhttps://www.blogger.com/profile/13511045845739028402noreply@blogger.com