Author |
Topic: 1,4,16,64... (Read 1474 times) |
|
ketanb
Newbie
Gender: 
Posts: 6
|
 |
1,4,16,64...
« on: Oct 24th, 2005, 6:57am » |
 Quote  Modify
|
Hi
many of us will be knowing the following problem:
one has to measure weights using a weight balance.Using how many minimum number of weights One can measure all the values from 1,2,3... N
provided:
1 it is allowed to put the weights only in one pan of the balance
2 it is allowed to put wts in both the pans
answer for part 1 is 1,2,4,8...
answer for part 2 is 1,3,9,27...(check it)
now here is part 3::
in above in 1 we used only " +" while in second we used "+" aswell as"-"
ex: 7 =4+2+1... if 1
=9-3+1...if 2
now we use one more operator namely hash
"#"
QUESTION IS :: DEFINE RULE FOR "#"
SO THAT USING ONLY 1,4,16,64,256....
AND OPERATORS "+, -, #" ONE IS ABLE TO DERIVE ALL NUMBERS
EX:
1=1
2=?
3=4-1
4=4
THEN 4#1=2: we define like this..
so Q is find relation for "a#b"
.....NOTE THAT
a#b is defined only if a,b are powers of 4
that is a#b#c is not defined provided
a#b itself is not power of 4(obv)
|
|
 IP Logged |
|
|
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13730
|
|
Re: 1,4,16,64...
« Reply #1 on: Oct 24th, 2005, 7:26am » |
Quote Modify
|
| hidden: |
I'll define # as [sqrt]
alternatively, if it has to be a binary operator, I'll define a # b as [sqrt]b
this reduces
1, 4, 16, 64, 256 ...
back to
1, 2, 4, 8, 16 ...
And we could already solve that with just +
|
|
| « Last Edit: Oct 24th, 2005, 7:26am by towr » |
IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
Grimbal
wu::riddles Moderator
Uberpuzzler
Gender:
Posts: 7527
|
|
Re: 1,4,16,64...
« Reply #2 on: Oct 24th, 2005, 8:25am » |
Quote Modify
|
I would define # = +2*
1 = 1
2 = #1 = +2*1
3 = 4-1
4 = 4
5 = 5+1
6 = 4#1 = 4+2*1
7 = #4-1 = +2*4-1
8 = #4 = +2*4
9 = #4+1 = +2*4+1
10 = #4#1 = +2*4+2*1
11 = 16-4-1
etc...
|
|
IP Logged |
|
|
|
ketanb
Newbie
Gender:
Posts: 6
|
|
Re: 1,4,16,64...
« Reply #3 on: Oct 24th, 2005, 9:09am » |
Quote Modify
|
sorry
i forgot to mention one condition
# is a function (of a AND b) such that
a#b is defined for a>b
and is inbetween a and b
|
|
IP Logged |
|
|
|
ChunkTug
Full Member
Gender:
Posts: 166
|
|
Re: 1,4,16,64...
« Reply #4 on: Oct 24th, 2005, 9:28am » |
Quote Modify
|
How about a#b = b+log4(a) then we can get rid of '+' and '-'
|
| « Last Edit: Oct 24th, 2005, 9:31am by ChunkTug » |
IP Logged |
|
|
|
Barukh
Uberpuzzler
Gender:
Posts: 2276
|
|
Re: 1,4,16,64...
« Reply #5 on: Oct 24th, 2005, 9:36am » |
Quote Modify
|
...or a#b = [sqrt](ab)?
|
|
IP Logged |
|
|
|
ketanb
Newbie
Gender:
Posts: 6
|
@Chunktug
how will u make nos like 40, 50.. some nos close to 64
by using ur formula?
remember that u have to minimize the no of numbers used
as stated in the first two versions of the problem
@Barukh
how will u make 22? using 1,4,16,64 only?
u cant use 256 here(if possible)'coz 64itself is greater than 22
like in our 1st 2 problems
to make wt say 41 u can use only 1,3,9,27,81
i have attached my answer:
|
|
IP Logged |
|
|
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13730
|
|
Re: 1,4,16,64...
« Reply #7 on: Oct 25th, 2005, 2:32am » |
Quote Modify
|
on Oct 25th, 2005, 1:04am, ketanb wrote:@Chunktug
how will u make nos like 40, 50.. some nos close to 64
by using ur formula?
remember that u have to minimize the no of numbers used
as stated in the first two versions of the problem |
|
He only ever needs to use two numbers.
say, 1 and 4n-1
4n-1#1 = 1 + log4 4n-1 = 1 + n-1= n
But I guess that's not what you mean.
|
| « Last Edit: Oct 25th, 2005, 2:41am by towr » |
IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
ketanb
Newbie
Gender:
Posts: 6
|
|
Re: 1,4,16,64...
« Reply #8 on: Oct 25th, 2005, 7:04am » |
Quote Modify
|
yes one can do that way
but
note that while doing this
to express "n" u need number which is nth in the sequencing which is>> the least number r such that4^r>n
as i said to weigh 41 u can go to 81 but not 243(just comparison)
because then in that case ur count of wts becomes
1,3,9,27,81,243=6 while u can work out just with 1st 5 wts in second case
|
|
IP Logged |
|
|
|
Grimbal
wu::riddles Moderator
Uberpuzzler
Gender:
Posts: 7527
|
|
Re: 1,4,16,64...
« Reply #9 on: Oct 25th, 2005, 7:30am » |
Quote Modify
|
I assume an implicit condition is that you can not use any one number 1,4,16,... more than once?
Under the new conditions, I can adjust my solution:
a#b = a-2*b
(or more precisely max(a-2*b,b+1) to meet the range conditions, even though I never use it where the max makes a difference).
Evaluation is always left to right.
1 = 1
2 = 4#1
3 = 4-1
4 = 4
5 = 4+1
6 = 16#4#1
7 = 16#4-1
8 = 16#4
9 = 16#4+1
10 = 16-4#1 (i.e. (16-4)#1)
11 = 16-4-1
12 = 16-4
13 = 16-4+1
14 = 16#1
15 = 16-1
16 = 16
17 = 16+1
18 = 16+4#1 (i.e. (16+4)#1)
etc...
But I come to think that if ( and ) are allowed, there must be much better solutions.
|
|
IP Logged |
|
|
|
Grimbal
wu::riddles Moderator
Uberpuzzler
Gender:
Posts: 7527
|
|
Re: 1,4,16,64...
« Reply #10 on: Oct 25th, 2005, 7:39am » |
Quote Modify
|
PS: I realize there evem is a simple physical device that implements this formula.
It is a standard 2-sided scale, but with one arm (on the negative side) extending further and hosting an additonal tray at twice the distance. There is one tray on the + side and 2 on the - side.
____________|______
| | ^ |
/_\ /_\ . /_\
|
| « Last Edit: Oct 25th, 2005, 7:40am by Grimbal » |
IP Logged |
|
|
|
ketanb
Newbie
Gender:
Posts: 6
|
|
Re: 1,4,16,64...
« Reply #11 on: Oct 26th, 2005, 1:25am » |
Quote Modify
|
@grimbal
yes
that is a solution provided a#b#c is allowed
but i u read the problem
i have stated that its not allowed
|
|
IP Logged |
|
|
|
Grimbal
wu::riddles Moderator
Uberpuzzler
Gender:
Posts: 7527
|
|
Re: 1,4,16,64...
« Reply #12 on: Oct 26th, 2005, 3:45pm » |
Quote Modify
|
Damn! You're right.
I'll have to think of something else.
Are brackets allowed?
If not, how is the priority?
is a#b+c#d = (a#b)+(c#d), ((a#b)+c)#d or (a#(b+c))#d) ?
i.e does it have lower, equal or higher priority than +, (provided the arguments could be made equal to a power of 2)?
|
|
IP Logged |
|
|
|
ketanb
Newbie
Gender:
Posts: 6
|
to be frank
i had not thought of that
any ways
i am attaching my answer here
|
|
IP Logged |
|
|
|
|
RIDDLES SITE
WRITE MATH!
Home 
Help 
Search 
Members 
Login 
Register
Reply
Notify of replies
Send Topic
Print
soln_to__problem.txt