bonanova Posted October 11, 2008 Report Share Posted October 11, 2008 (edited) I'll make it short and sweet tonight, Alex said as he greeted the boys last night at Morty's, because yer gonna need every second of time you've got to figure this one out. That turned a few heads, but only Ian and Jamie bothered to sit down and listen. There's this puzzle contest I read about, and it has one of those sequences of numbers with one number missing. It's a shame to waste it on buzzards like yerselves, but I'll let you have a go at it, anyway. Here it is: 35, 46, 60, z, 120, 180, 280, 450, 744, 1260 Edit: '46' is incorrect. Here's the correct sequence. Thanks to Grey Cells for pointing it out. 35, 45, 60, z, 120, 180, 280, 450, 744, 1260 As Ian and Jamie stared at the numbers Alex had scrawled out in his dirty red ink, Alex continued, Kind of interesting that one of them doesn't end in 5 or 0 isn't it? They both nodded, but Jamie was about to venture a guess anyway. Well, there's an even bigger monkey wrench in the works, Alex concluded, and it's this: z is not an integer. At that, Ian and Jamie rolled their eyes and decided to go play darts. That leaves you to solve the puzzle. Enjoy! Edited October 13, 2008 by bonanova Correct one of the terms Quote Link to comment Share on other sites More sharing options...
0 grey cells Posted October 11, 2008 Report Share Posted October 11, 2008 79.5..? Did some very basic calculation, so I could be wrong by quite a margin. But the answer should be 80ish I think. EDIT: By the way, there are 2 numbers not ending with 0 or 5. Is one of them 45 or 745? Quote Link to comment Share on other sites More sharing options...
0 Izzy Posted October 11, 2008 Report Share Posted October 11, 2008 (edited) Okay, so I figured the only way to get a number ending in a decimal using the numbers in the sequence would be by dividing. So I divided each number by its predecessor, and got a bit of a pattern. 46/35 = 1.31428571 60/46 = 1.30434783 z/60 = ___________ 120/z = __________ 180/120 = 1.5 280/180 = 1.5555... 450/280 = 1.60714286 744/450 = 1.65333... 1260/744 = 1.69354839 So, not completely a pattern, but it does increase a bit each time. Since z is the 4th number, I thought if we average the number you get from dividing on either side of z, you'd know what to divide 120 by to get z. So... (1.31428571 + 1.69354839)/2 = 1.46071428 z = 82.1516728? *edit* Typo Edited October 11, 2008 by Izzy Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted October 11, 2008 Author Report Share Posted October 11, 2008 The sequence has been corrected - thanks Grey Cells for calling attention to it. 35, 45, 60, z, 120, 180, 280, 450, 744, 1260 Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted October 11, 2008 Author Report Share Posted October 11, 2008 Okay, so I figured the only way to get a number ending in a decimal using the numbers in the sequence would be by dividing. So I divided each number by its predecessor, and got a bit of a pattern. 46/35 = 1.31428571 60/46 = 1.30434783 z/60 = ___________ 120/z = __________ 180/120 = 1.5 280/180 = 1.5555... 450/280 = 1.60714286 744/450 = 1.65333... 1260/744 = 1.69354839 So, not completely a pattern, but it does increase a bit each time. Since z is the 4th number, I thought if we average the number you get from dividing on either side of z, you'd know what to divide 120 by to get z. So... (1.31428571 + 1.69354839)/2 = 1.46071428 z = 82.1516728?*edit* Typo Izzy, you're close, of course. [note that 46 was written incorrectly: it's actually 45]. And I'll stipulate that the answer lies on some kind of smooth curve. So let me suggest the solution should either be exact to 5 decimal places [e.g. XX.YYYYY] where not all the X's or Y's are [necessarily] equal, and of course some of the trailing Y's could be zero, or a formula, say, that generates the numbers. Quote Link to comment Share on other sites More sharing options...
0 Izzy Posted October 11, 2008 Report Share Posted October 11, 2008 So if I plugged in 45 instead, would I be right? Or is this exponentials and other way over my head maths stuff like my friend said? Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted October 12, 2008 Author Report Share Posted October 12, 2008 So if I plugged in 45 instead, would I be right? Or is this exponentials and other way over my head maths stuff like my friend said? That's a fair question. What I'll say is that I asked for something more exact than anything you could get from curve fitting so you wouldn't try that way to solve it Next I'll say there is an expression, not an overly difficult one, that generates all the numbers in the OP, including the unknown one. The expression is interesting because it generates this string of integer values with an irrational number right in the middle. If you're wondering, even knowing the missing number - and you're in the ball park - may not help you find the expression. That's general information. Now here are two clues:consecutive integer values: it's f(n) where the sequence in the OP comes from consecutive integer values of n. a particular value of n that might generate a distinctive result. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 12, 2008 Report Share Posted October 12, 2008 75 is the answer Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted October 12, 2008 Author Report Share Posted October 12, 2008 75 is the answer Hi Ian, and welcome to the Den. The puzzle states that the answer isn't a whole number, so that isn't it, sorry. Hint: putting answers into spoilers hides them for others who want to solve on their own. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 12, 2008 Report Share Posted October 12, 2008 That's a fair question. What I'll say is that I asked for something more exact than anything you could get from curve fitting so you wouldn't try that way to solve it Next I'll say there is an expression, not an overly difficult one, that generates all the numbers in the OP, including the unknown one. The expression is interesting because it generates this string of integer values with an irrational number right in the middle. If you're wondering, even knowing the missing number - and you're in the ball park - may not help you find the expression. That's general information. Now here are two clues:consecutive integer values: it's f(n) where the sequence in the OP comes from consecutive integer values of n. a particular value of n that might generate a distinctive result. I plotted the sequence, and it just has to have a^n in it. The second hint indicates that z=f(0). I don't know what to do to a^n to give an irrational result at f(0), and integer values elsewhere though... Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 12, 2008 Report Share Posted October 12, 2008 I plotted the sequence, and it just has to have a^n in it. The second hint indicates that z=f(0). I don't know what to do to a^n to give an irrational result at f(0), and integer values elsewhere though... (21/2)2^n gives an irrational result at n = 0 and integer results elsewhere, but that is a double exponential function. Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted October 13, 2008 Author Report Share Posted October 13, 2008 I plotted the sequence, and it just has to have a^n in it. The second hint indicates that z=f(0). I don't know what to do to a^n to give an irrational result at f(0), and integer values elsewhere though... ... that the exp​ression has a limit operation. That is, f(n) = lim{x->n}f(x) for consecutive integer values of n, where f(x) is continuous except at the special value. So for f = {35, 45, 60, 120, 180, 280, 744, 1260} the limit isn't needed; at the special value, f isn't continuous; but its limit exists. Quote Link to comment Share on other sites More sharing options...
0 Izzy Posted October 13, 2008 Report Share Posted October 13, 2008 (edited) So I asked my maths teacher. She didn't know (..which I kind of expected). My science teacher is e-mailing one of her friends about it though! I _will not_ let this sequence go unsolved! *sulks at not being able to solve it due to a lack of math skills* Like, I know how to do f(x), if {x->n} is similar to {p->q}, then I know that as well, but what's the limit bit? Edited October 13, 2008 by Izzy Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 13, 2008 Report Share Posted October 13, 2008 ... that the exp​ression has a limit operation. That is, f(n) = lim{x->n}f(x) for consecutive integer values of n, where f(x) is continuous except at the special value. So for f = {35, 45, 60, 120, 180, 280, 744, 1260} the limit isn't needed; at the special value, f isn't continuous; but its limit exists. Sometimes you make a wild guess and get lucky. 120 ln(2) Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted October 13, 2008 Author Report Share Posted October 13, 2008 Sometimes you make a wild guess and get lucky. 120 ln(2) Right. The function, tho not horribly complex, isn't totally obvious. Or given that you know the answer, maybe you can construct it. Quote Link to comment Share on other sites More sharing options...
0 Izzy Posted October 13, 2008 Report Share Posted October 13, 2008 Okay, so I gave the sequence and formula to my science teacher who forwarded it to his assistant. This is what he got: 120 ln(2) = 83.1776617 Gah I was close! Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted October 13, 2008 Author Report Share Posted October 13, 2008 Okay, so I gave the sequence and formula to my science teacher who forwarded it to his assistant. This is what he got: 120 ln(2) = 83.1776617 Gah I was close! Did he also come up with the formula by chance ? Quote Link to comment Share on other sites More sharing options...
0 Izzy Posted October 13, 2008 Report Share Posted October 13, 2008 Did he also come up with the formula by chance ? Well, the f(n) = lim(x->n)f(x) was e-mailed to him. So yes, unless you mean another formula. Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted October 13, 2008 Author Report Share Posted October 13, 2008 Well, the f(n) = lim(x->n)f(x) was e-mailed to him. So yes, unless you mean another formula. f(x) - what is it? Quote Link to comment Share on other sites More sharing options...
0 Izzy Posted October 13, 2008 Report Share Posted October 13, 2008 No idea. I'll ask though. Gah, when I subtract my answer from his, I was *soooo* close! Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 13, 2008 Report Share Posted October 13, 2008 f(x) - what is it? 120/x * (2^x - 1) Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted October 14, 2008 Author Report Share Posted October 14, 2008 120/x * (2^x - 1) Jackpot. Quote Link to comment Share on other sites More sharing options...
Question
bonanova
I'll make it short and sweet tonight, Alex said as
he greeted the boys last night at Morty's, because
yer gonna need every second of time you've got
to figure this one out.
That turned a few heads, but only Ian and Jamie
bothered to sit down and listen. There's this puzzle
contest I read about, and it has one of those sequences
of numbers with one number missing. It's a shame to
waste it on buzzards like yerselves, but I'll let you have
a go at it, anyway. Here it is:
35, 46, 60, z, 120, 180, 280, 450, 744, 1260
Edit: '46' is incorrect. Here's the correct sequence.
Thanks to Grey Cells for pointing it out.
35, 45, 60, z, 120, 180, 280, 450, 744, 1260
As Ian and Jamie stared at the numbers Alex had
scrawled out in his dirty red ink, Alex continued,
Kind of interesting that one of them doesn't end in 5 or 0
isn't it? They both nodded, but Jamie was about
to venture a guess anyway.
Well, there's an even bigger monkey wrench in the
works, Alex concluded, and it's this: z is not an integer.
At that, Ian and Jamie rolled their eyes and decided to
go play darts.
That leaves you to solve the puzzle.
Enjoy!
Edited by bonanovaCorrect one of the terms
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