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Topic: Rounded Roots (Read 372 times) |
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Sir Col
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Rounded Roots
« on: Jun 28th, 2003, 3:16am » |
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A special number machine will accept any natural number, find its square root and then output the result rounded to the nearest whole number.
For example,
32 — square root —> 5.656... — rounded —> 6
1. If 6 is the output, what is the set of possible input numbers?
2. For a given output, n, find the set of possible input numbers.
3. Prove your result.
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TenaliRaman
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Re: Rounded Roots
« Reply #1 on: Jun 28th, 2003, 8:32am » |
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1> Inputs={x|31<=x<=42}
2> Inputs={x|((n-1)2+k)<=x<=(n2+k)}
where k = floor((4n-3)/4)+1
3>
note that all such number will be greater than (n-1)2
our inputs will be simply determined by an offset from (n-1)2.Take this offset as k,
Then it is easy to note that,
sqrt((n-1)2+k)>(n+(n-1))/2
solving this for k gives the answer.
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| « Last Edit: Jun 28th, 2003, 8:33am by TenaliRaman » |
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Sir Col
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Re: Rounded Roots
« Reply #2 on: Jun 28th, 2003, 9:36am » |
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Nice solution, TenaliRaman. What a clever and original approach too. Having done the hardest part, you might like to simplify it, as you don't need the floor function. Consider: sqr((n–1)2+k)>=n–0.5 
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| « Last Edit: Jun 28th, 2003, 9:41am by Sir Col » |
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TenaliRaman
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Re: Rounded Roots
« Reply #3 on: Jun 28th, 2003, 10:04am » |
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D'oh!! ofcourse !!!
k = floor((4n-3)/4)+1 = floor(n-3/4)+1 = n - 1 + 1 = n
P.S-> yes ofcourse my avatar.(should've noticed that!!)
This P.S should've been in the other topic 
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| « Last Edit: Jun 28th, 2003, 10:07am by TenaliRaman » |
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Self discovery comes when a man measures himself against an obstacle - Antoine de Saint Exupery
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Sir Col
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Posts: 1825
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Re: Rounded Roots
« Reply #4 on: Jun 28th, 2003, 10:12am » |
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What if the number machine cube roots?
(I've not solved this myself yet)
P.S. Oh well, it'll add a bit of intrigue (and confusion) for anyone who's not read the other topic – especially as they don't know what the other topic or the question was.
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TenaliRaman
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Re: Rounded Roots
« Reply #5 on: Jun 28th, 2003, 10:54am » |
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i followed the same approach as above for the cube root which gives,
k = floor( (12n2-18n+7)/8 )+1
Simplification of this though seems difficult.
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