tag:blogger.com,1999:blog-2143560514333540409.post2896251169978161980..comments2022-10-27T22:15:02.786-04:00Comments on Counter Rhythms: Can you solve Papa's puzzles? by David GilfixDavid Gilfixhttp://www.blogger.com/profile/04275956364860425006noreply@blogger.comBlogger12125tag:blogger.com,1999:blog-2143560514333540409.post-31051631748388480712013-11-30T22:11:36.220-05:002013-11-30T22:11:36.220-05:00David -
Thanks for the puzzles.
I wanted to add o...David -<br />Thanks for the puzzles.<br /><br />I wanted to add one note to the puzzle of the children's ages. School age isn't really the key factor here. The choices for ages were (see #8):<br />1,2,20<br />1,4,10<br />1,5,8<br />2,2,10<br />2,4,5<br />plus (theoretically) 1,1,40.<br /><br />The clue is that Bill couldn't tell the ages just by knowing the product and sum. (Bill presumably knows the house number even though we don't; after all, he's at Bert's house.) Only 1,5,8 and 2,2,10 give the same sum, so the house number must be 14 if Bill couldn't guess the ages. However, when Bert mentions "the two elder kids" that eliminates the 2,2,10 combination, leaving 1,5,8.Howard Waldmannoreply@blogger.comtag:blogger.com,1999:blog-2143560514333540409.post-69289744243757152682011-01-18T22:34:05.510-05:002011-01-18T22:34:05.510-05:00SPOILER ALERT!
ANSWERS TO FOLLOW!
The responses i...SPOILER ALERT!<br />ANSWERS TO FOLLOW!<br /><br />The responses in the comments section confirmed what I already knew; I have some very math-smart readers! Instead of providing my answers to each question, I will refer to answers and explanations in this comment section (since they are way better than anything I could come up with).<br /><br />There are 11 comments (including this one); I will refer each comment by its number order.<br />(Incidentally, you are reading comment #11)<br /><br />Question 1, “How long is the Shelf?” - ssancetta (#5) and anonymous (#8) gave clear explanations for the problem as most people would read it.<br /><br />Mathematician Scott Axelrod (#2) noticed an ambiguity in the wording of the problem (like a true mathematician), reworded the problem, and then solved the reworded problem, providing a different answer than #5 and #8. I am sure Scott is right because this kind of problem was already too easy for him in elementary school.<br /><br />Question 2, “How old are Bert’s children?” - Of all the correct answers, anonymous (#8) gave the most colorful explanation.<br /><br />Question 3, “Hanged or drowned?” - Two different solutions. Steve (#3) and ssancetta (#6).<br /><br />Question 4, “Straightforward math?” - I couldn’t figure this out, until my math-smart Russian neighbor explained it to me (yes, sometimes stereotypes are accurate!). Anyway, Scott Axelrod (#4) explains the answer.<br /><br />Question 5, “Measuring the bookworm.” - Anonymous (#9).<br /><br />Question 6, “Who done it?” - Anonymous (#10), who plays a mean game of chess, got the same answer as my jack-of-all-trades cousin and I. Interestingly, this is the only puzzle for which we couldn’t find Papa’s answer.David Gilfixhttps://www.blogger.com/profile/17909283071066269141noreply@blogger.comtag:blogger.com,1999:blog-2143560514333540409.post-48306431966170068092010-08-25T15:47:24.767-04:002010-08-25T15:47:24.767-04:00Puzzle #6
This is actually very easy just some si...Puzzle #6<br /><br />This is actually very easy just some simple logic which they don't teach at school. I leave out some details but you should be able to follow. Let me know before posting. <br /><br />Consider<br /><br />Statement:<br />O1 - O'Neil statement 1<br />O2<br />O3<br /><br />R1<br />R2<br />R3<br /><br />W1<br />W2<br />W3<br /><br />R3 and O3 say the same thing<br /><br />If both are correct then W3 is false, which means W1 and W2 are true. <br /><br />If both are false (R3 and O3) then either R2 or O2 is false which cannot be by the rules.<br /><br />So if W3 is false then Watts did not do it and he was sitting next to O'Neil<br /><br />This makes R1 false, which means the waiter did not do it by R3<br /><br />This makes O2 false.<br /><br />So waiter did not do it, Watts did not do it W1, Rogers did not do it O1.<br /><br />This leaves only O'Neil.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-2143560514333540409.post-1407467349788191082010-08-25T11:27:18.123-04:002010-08-25T11:27:18.123-04:00Hang my head in shame... I normally pride myself i...Hang my head in shame... I normally pride myself in mastery of patterns and sequences, but I rushed on number 4 (whilst working at night), and didn't test my work... this sort of oversight is what leads to satellites plunging into deep space, if your line of work is avionics for NASA. So, my incorrect response came to me as I was driving into Cambridge this morning. The correct answer is:<br /><br />(2*3*4*5*6*7*8*9*10)+1=3628801<br /><br />The remainder piece is obvious, because the number is one greater than a number which is divisible by all factors. The divisible by 11 piece, with no remainder, is only slightly more obtuse.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-2143560514333540409.post-60573811175640799712010-08-24T23:33:36.988-04:002010-08-24T23:33:36.988-04:00Puzzle 1:
x=thickness of book, in inches
y=number...Puzzle 1:<br /><br />x=thickness of book, in inches<br />y=number of books<br /><br />first case:<br />xy=(x-1)*(y+6)<br />xy=xy+6x-y-6<br />6x=y+6<br /><br />second case:<br />xy=(x+1)*(y-3)<br />xy=xy-3x+y-3<br />3x=y-3<br />6x=2y-6<br /><br />combining:<br />6x=y+6=2y-6<br />y=12<br />6x=12+6<br />x=3<br /><br />bookshelf=xy=36 inches<br /><br />Puzzle 2:<br /><br />combinations yielding 40, when multiplied:<br /><br />20*2*1 (20 is too old for school, except in parts of Kentucky)<br />10*2*2 (2 is too young for school, unless your dad is Stephen Hawking)<br />10*4*1 (4 is too young for school)<br />5*4*2 (2 is too young for school)<br />So, it must be the only other unique combination, with two ages being school ages:<br />8*5*1<br /><br />Puzzle 3:<br /><br />No answer... logic is for those who can't think straight.<br /><br />Puzzle 4:<br /><br />(2*3*4*5*6*7*8*9)-1=362879<br /><br />Puzzle 5: (I admit that after all these years I had to look up the formula for a finite series)<br /><br />x=pages per day<br />sum of a + kd, for k=0 to n-1, where a=1/4x,d=1,n=31<br />(n/2)*(2a+(n-1)d)=(31/2)*(2*(1/4x)+(31-1)*1)=31/4x+1860<br /><br />so,<br />31x=31/4x+465<br />124x-31x=1860<br />93x=1860<br />x=20<br /><br />pages in book=20*31=620<br /><br /><br /><br />Puzzle 6:<br />See answer for puzzle 3.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-2143560514333540409.post-9343226457328344922010-08-24T15:50:43.370-04:002010-08-24T15:50:43.370-04:00Puzzle 2: I also get 8, 5 and 1... tho I suppose i...Puzzle 2: I also get 8, 5 and 1... tho I suppose it could be 4, 5 and 2. Depends on whetehr you want your 4 year old walking home from kindergarten.ssancettahttps://www.blogger.com/profile/05053817982786476380noreply@blogger.comtag:blogger.com,1999:blog-2143560514333540409.post-25044956001935656632010-08-24T15:47:03.874-04:002010-08-24T15:47:03.874-04:00Puzzle 3 - Stymie the judges with this: "Thi...Puzzle 3 - Stymie the judges with this: "This statement is false"ssancettahttps://www.blogger.com/profile/05053817982786476380noreply@blogger.comtag:blogger.com,1999:blog-2143560514333540409.post-84703385856993008262010-08-24T15:05:06.152-04:002010-08-24T15:05:06.152-04:00b is number of books
w is width of a book
the wid...b is number of books<br />w is width of a book<br /><br />the width of the shelf is thus b * w<br />the width of the shelf is ALSO (b+6) * (w - 1)<br />the width of the shelf is ALSO (b-3) * (w + 1)<br /><br /> so<br />b* w = (b+6) * (w - 1)<br />w/(w-1) = (b+6)/b = 1 + 6/b<br />w/(w-1) - 1 = 6/b<br /><br /> and<br />b * w = (b-3) * (w + 1)<br />w/(w+1) = (b-3)/b = 1 - 3/b<br />1 - w/(w+1) = 3/b<br />2 - 2w/(w+1) = 6/b<br /><br /> combining the previous 2 results, with 2 expressions in w both equal to 6/b<br />w/(w-1) - 1 = 2 - 2w/(w+1)<br /><br /> multiplying each side by (w-1)*(w+1)<br />w(w+1) - (w-1)(w+1) = 2(w-1)(w+1) - 2w(w-1)<br />w*w + w - (w*w - 1) = 2w*w - 2 - (2w*w - 2w)<br />w + 1 = 2w - 2<br />w = 3<br /><br /> substituting 3 for w in the first equation<br /><br />b*3 = (b+6)*(3-1)<br />b*3 = b*2 + 12<br />b = 12<br /><br />so the shelf width = 3 * 12 = 36ssancettahttps://www.blogger.com/profile/05053817982786476380noreply@blogger.comtag:blogger.com,1999:blog-2143560514333540409.post-928679050378918612010-08-23T22:24:10.715-04:002010-08-23T22:24:10.715-04:00I should have said in my previous post that your g...I should have said in my previous post that your grandpa seems fascinating and had a good selection of problems. (Also I should have given a spoiler alert that my response was a solution to Puzzle 1.)<br /><br />So now I give a spoiler alert that the following is my solution for Puzzle 4.<br /><br /><br />The answer is of form 1+ x*(5*7*8*9) for x chosen so that answer is a mutiple of 11. For this to be the case, the remainder of x*(5*7*8*9) divided by 11 must be -1. Since the remainder of 5*7*8*9 divided by 11 is 1, we can take x=-1. For x=-1 we get the answer -2519. We can add any multiple of 11 to x. For x=10, the we get the answer 25201.Scott Axelrodhttps://www.blogger.com/profile/08031127269601065282noreply@blogger.comtag:blogger.com,1999:blog-2143560514333540409.post-32893258036354802422010-08-23T21:25:00.153-04:002010-08-23T21:25:00.153-04:002: 8, 5, and 1 it seems.
3: "I will be hanged...2: 8, 5, and 1 it seems.<br />3: "I will be hanged."<br />Will try to look at the rest when I have more time.<br />Thanks - to you and your grandfather. If he was around now he probably would be blogging.<br />SteveStevenoreply@blogger.comtag:blogger.com,1999:blog-2143560514333540409.post-20467871197970437322010-08-23T21:08:53.733-04:002010-08-23T21:08:53.733-04:00I would reword the problem a little:
A shelf is...I would reword the problem a little:<br /> A shelf is exactly filled with books of equal thickness.<br /> If the books were 1" thinner, the shelf would accommodate 6 more books (BUT NOT SEVEN).<br /> If the books were 1" thicker, then there would be no room for 3 books (BUT IT COULD FIT TWO).<br /> How many books are on the shelf?<br /><br /><br />Answer: 4 books are on the shelf. The books are between 11/7 and 10/6 inches long. The<br />shelf is between 44/7 and 40/6 inches long.<br /><br />Let N be the number of books, T be their thickness, and L=N*T be the length of the shelf.<br />The 1" thinner condition tells us that:<br /> (N+6)*(T-1) <= L=N*T < (N+7)*(T-1)<br /><br />This can be rewritten as:<br /> [*] 1 + N/7 < T <= 1 + N/6<br /><br />The 1" thicker condition tells us that:<br /> 2*(T+1) <= L=N*T < 3*(T+1)<br /><br />The can be rewritten as:<br /> [**] 2/(N-2) <= T < 3/(N-3)<br /><br />There can't be 3 or fewer books on the shelf, because then the upper bound in [*] tell us that T is smaller (or equal to) 1 1/2 and the<br />lower bound from [**] says that T is bigger (or equal) to 2.<br />Similarly, there can't be 5 or more books on the shelf, because then the lower bound in [*] tells us that T is larger than 12/7 and the upper bound in [**] tells us that T is smaller than 3/2.<br /><br />So there must be 4 books on the shelf. Condition [*] tells us that 11/7 < T <= 10/6. Condition [*] tells us that 1/2 <= T < 3.<br /><br /><br />-----<br /><br /><br />Here is a variant of the above problem suitable for interest elementary schoolers.<br /><br />A shelf is exactly filled with books of equal thickness. The thickness is some whole number (no fractions) of inches.<br />If the books were 1" thinner, the shelf would accommodate 2 more books (exactly, with no space left over).<br />If the books were 1" thicker, the shelf would accomodate one fewer books (exactly, with no space left over).<br /><br />How many books are on the shelf? How thick are the books? How long is the shelf?Scott Axelrodhttps://www.blogger.com/profile/08031127269601065282noreply@blogger.comtag:blogger.com,1999:blog-2143560514333540409.post-67227791226718177602010-08-23T17:39:47.899-04:002010-08-23T17:39:47.899-04:00Brilliant stuff, David. Sounds like your grandfat...Brilliant stuff, David. Sounds like your grandfather was a wonderful man. I need to spend some time on these!!Lesternoreply@blogger.com