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Topic: My broken calculator (Read 602 times) |
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BNC
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My broken calculator
« on: Sep 27th, 2003, 3:09pm » |
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My calculator just broke down. Almost nothing works. In fact, only 3 functions still work:
Function A doubles the current result (res -> 2*res)
Function B adds one to the current result (res -> res + 1)
Function C clears the result (0 -> res)
I just pressed C, and the result is 0.
1. What is the minimal number of actions required to get a result of 100?
2. What is the maximal number that is still smaller than 100, which fulfils the minimax condition: the minimal number of actions required to get it is maximal?
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| « Last Edit: Sep 27th, 2003, 3:11pm by BNC » |
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How about supercalifragilisticexpialidociouspuzzler [Towr, 2007]
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Lightboxes
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Re: My broken calculator
« Reply #1 on: Sep 27th, 2003, 11:44pm » |
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I'm confused about this one? It's zero and you hit the C button to get res?
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I hope I don't sound stupid:
1) ::Hit B 3 times, then hit A 3 times, then hit B 1 time then hit A 2 times.
9 times total
2) And if I understood this one correctly...I got 97:
Hit B 1 time, A 1 time, B 1 time, A 5 times, hit B 1 time.
9 times total.
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| « Last Edit: Sep 27th, 2003, 11:52pm by Lightboxes » |
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ThudnBlunder
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Re: My broken calculator
« Reply #2 on: Sep 28th, 2003, 12:07am » |
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Quote:| I'm confused about this one? It's zero and you hit the C button to get res? |
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No, when 'res' (result) is showing you press C to clear so that it now shows zero.
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BNC
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Re: My broken calculator
« Reply #3 on: Sep 28th, 2003, 12:48am » |
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on Sep 27th, 2003, 11:44pm, Lightboxes wrote:I'm confused about this one? It's zero and you hit the C button to get res?
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Answered by T&B.
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I hope I don't sound stupid:
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Never!
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Correct. Now generalize for any number.
hint: A certain observation is the base for an easy answer
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2) And if I understood this one correctly...I got 97:
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Let me try to explain. Take any number in the range [1,100]. For every number n, there exist a minimal number of actions required to achieve that n -- call it m(n). Out of the set {m(n), n=1..100}, we define the maximum M=max(m(n), n=1..100). There are a few numbers that fulfill m(n)=M. We're looking for N, that is the maximal number < 100 with m(N)=M.
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| « Last Edit: Sep 28th, 2003, 12:56am by BNC » |
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How about supercalifragilisticexpialidociouspuzzler [Towr, 2007]
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Lightboxes
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Re: My broken calculator
« Reply #4 on: Sep 28th, 2003, 8:20pm » |
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I would generalize by saying if n equals an even number, then divide by two and continue just before the result is a number with a decimal nn.5, then subract one, then divide by 2's again...then stop right before a result equals nn.5, then subract one...etc.
::So...1 would a solution to 2)? and 2, 3? 3 being the max
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| « Last Edit: Sep 28th, 2003, 8:34pm by Lightboxes » |
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wowbagger
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Re: My broken calculator
« Reply #5 on: Sep 29th, 2003, 3:47am » |
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on Sep 28th, 2003, 12:48am, BNC wrote:
Correct. Now generalize for any number.
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So we want to get a number n using functions A (*2) and B (+1). First, write n in base 2, e.g. (100)10 = (1100100)2. Now, for every 1 that shows up in the base-2 representation of n, we have to press B once. Function A shifts the ones and zeros one place to the left, inserting a zero at the end. Let's say n has d digits in base 2, o of which are ones (d=7, o=3 for n=100). For every digit (starting from the second), we have to press A once. In the case n = 100, we use the sequence BABAAABAA - which is different from Lightboxes's solution, but probably more easily generalized.
For general n, the minimal number of actions is o + d - 1.
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| « Last Edit: Sep 29th, 2003, 3:51am by wowbagger » |
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wowbagger
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Re: My broken calculator
« Reply #6 on: Sep 29th, 2003, 4:09am » |
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I'll try to clarify BNC's explanation on part 2 using an example.
Note that the hidden part below is a spoiler for the generalized case of question 1, and possibly a hint for part 2.
BNC: "Take any number in the range [1,100]."
I restrict this example to [1,10].
BNC: "For every number n, there exist a minimal number of actions required to achieve that n -- call it m(n)."
For the calculation of m(n) from the binary representation of n, see my last post.
n (decimal) n (binary) m(n)
1 1 1
2 10 2
3 11 3
4 100 3
5 101 4
6 110 4
7 111 5
8 1000 4
9 1001 5
10 1010 5
BNC: "Out of the set {m(n), n=1..100}, we define the maximum M=max(m(n), n=1..100)."
In my example, M = 5, because this is the maximum that shows up in the third column above.
BNC: "There are a few numbers that fulfill m(n)=M."
These are {7, 9, 10}.
BNC: "We're looking for N, that is the maximal number < 100 with m(N)=M."
The maximum of {7, 9, 10} is, of course, 10. In this respect, my example is rather boring. If you extend this to include numbers up to 100, however, it gets more interesting: 63, for example, needs more operations than 100.
Hope this helps to understand the mathematical formulation of part 2 BNC gave.
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| « Last Edit: Sep 29th, 2003, 4:13am by wowbagger » |
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Lightboxes' Clone
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I understand it now, now that I look back at BNC's explanation. I must have been too tired since I came from work yesterday. And thanks wowbagger.
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