I made a simple Fibonacci sequence calculator for the first 22 terms:
i=1
n=0
while i<=20000:
i = i + n
n = i - n
print(i)
Looks like the result is correct
1
2
3
5
8
13
21
34
55
89
144
233
377
610
987
1597
2584
4181
6765
10946
17711
28657
but I can't seem to find similar code anywhere online. I think that's a big red flag. Can someone tell me what is wrong here? Is this inefficient code?
No, that code is fine. The probable reason you can't find similar code online is that it's unusual to use the subtraction operator in Fibonacci, which is a purely additive function, tn = tn-2 + tn-1.
It works, of course, since addition/subtraction is both commutative and associative, meaning that order and grouping of terms is unimportant:
i = i + n # iNew = iOld + nOld
n = i - n # nNew = (iNew) - nOld
# = (iOld + nOld) - nOld
# = iOld + (nOld - nOld)
# = iOld + (0)
# = iOld
Use of subtraction allows you to bypass needing a third variable, which would be something like this in a lesser language than Python:
nextN = i + n
i = n
n = nextN
In Python, you don't actually need that since you can use tuple assignment such as:
(n, i) = (i, n + i)
With that, everything on the right of the = is evaluated before any assignments to the left.
It's an unusual way to do it, but it's correct. Your lines:
i = i + n
n = i - n
are the same as doing:
new_i = i + n
n = i
i = new_i
or,
i, n = i + n, i
which would be the usual way in Python.
The R ppoints function is described as:
Ordinates for Probability Plotting
Description:
Generates the sequence of probability points ‘(1:m - a)/(m +
(1-a)-a)’ where ‘m’ is either ‘n’, if ‘length(n)==1’, or
‘length(n)’.
Usage:
ppoints(n, a = ifelse(n <= 10, 3/8, 1/2))
...
I've been trying to replicate this function in python and I have a couple of doubts.
1- The first m in (1:m - a)/(m + (1-a)-a) is always an integer: int(n) (ie: the integer of n) if length(n)==1 and length(n) otherwise.
2- The second m in the same equation is NOT an integer if length(n)==1 (it assumes the real value of n) and it IS an integer (length(n)) otherwise.
3- The n in a = ifelse(n <= 10, 3/8, 1/2) is the real number n if length(n)==1 and the integer length(n) otherwise.
This points are not made clear at all in the description and I'd very much appreciate if someone could confirm that this is the case.
Add
Well this was initially posted at https://stats.stackexchange.com/ because I was hoping to get the input of staticians who work with the ppoints function. Since it has been migrated here, I'll paste below the function I wrote to replicate ppoints in python. I've tested it and both seem to give back the same results, but I'd be great if someone could clarify the points made above because they are not made at all clear by the function's description.
def ppoints(vector):
'''
Mimics R's function 'ppoints'.
'''
m_range = int(vector[0]) if len(vector)==1 else len(vector)
n = vector[0] if len(vector)==1 else len(vector)
a = 3./8. if n <= 10 else 1./2
m_value = n if len(vector)==1 else m_range
pp_list = [((m+1)-a)/(m_value+(1-a)-a) for m in range(m_range)]
return pp_list
I would implement this with numpy:
import numpy as np
def ppoints(n, a):
""" numpy analogue or `R`'s `ppoints` function
see details at http://stat.ethz.ch/R-manual/R-patched/library/stats/html/ppoints.html
:param n: array type or number"""
try:
n = np.float(len(n))
except TypeError:
n = np.float(n)
return (np.arange(n) + 1 - a)/(n + 1 - 2*a)
Sample output:
>>> ppoints(5, 1./2)
array([ 0.1, 0.3, 0.5, 0.7, 0.9])
>>> ppoints(5, 1./4)
array([ 0.13636364, 0.31818182, 0.5 , 0.68181818, 0.86363636])
>>> n = 10
>>> a = 3./8. if n <= 10 else 1./2
>>> ppoints(n, a)
array([ 0.06097561, 0.15853659, 0.25609756, 0.35365854, 0.45121951,
0.54878049, 0.64634146, 0.74390244, 0.84146341, 0.93902439])
One can use R fiddle to test implementation.
I got one puzzle and I want to solve it using Python.
Puzzle:
A merchant has a 40 kg weight which he used in his shop. Once, it fell
from his hands and was broken into 4 pieces. But surprisingly, now he
can weigh any weight between 1 kg to 40 kg with the combination of
these 4 pieces.
So question is, what are weights of those 4 pieces?
Now I wanted to solve this in Python.
The only constraint i got from the puzzle is that sum of 4 pieces is 40. With that I could filter all the set of 4 values whose sum is 40.
import itertools as it
weight = 40
full = range(1,41)
comb = [x for x in it.combinations(full,4) if sum(x)==40]
length of comb = 297
Now I need to check each set of values in comb and try all the combination of operations.
Eg if (a,b,c,d) is the first set of values in comb, I need to check a,b,c,d,a+b,a-b, .................a+b+c-d,a-b+c+d........ and so on.
I tried a lot, but i am stuck at this stage, ie how to check all these combination of calculations to each set of 4 values.
Question :
1) I think i need to get a list all possible combination of [a,b,c,d] and [+,-].
2) does anyone have a better idea and tell me how to go forward from here?
Also, I want to do it completely without help of any external libraries, need to use only standard libraries of python.
EDIT : Sorry for the late info. Its answer is (1,3,9,27), which I found a few years back. I have checked and verified the answer.
EDIT : At present, fraxel's answer works perfect with time = 0.16 ms. A better and faster approach is always welcome.
Regards
ARK
Earlier walk-through anwswer:
We know a*A + b*B + c*C + d*D = x for all x between 0 and 40, and a, b, c, d are confined to -1, 0, 1. Clearly A + B + C + D = 40. The next case is x = 39, so clearly the smallest move is to remove an element (it is the only possible move that could result in successfully balancing against 39):
A + B + C = 39, so D = 1, by neccessity.
next:
A + B + C - D = 38
next:
A + B + D = 37, so C = 3
then:
A + B = 36
then:
A + B - D = 35
A + B - C + D = 34
A + B - C = 33
A + B - C - D = 32
A + C + D = 31, so A = 9
Therefore B = 27
So the weights are 1, 3, 9, 27
Really this can be deduced immediately from the fact that they must all be multiples of 3.
Interesting Update:
So here is some python code to find a minimum set of weights for any dropped weight that will span the space:
def find_weights(W):
weights = []
i = 0
while sum(weights) < W:
weights.append(3 ** i)
i += 1
weights.pop()
weights.append(W - sum(weights))
return weights
print find_weights(40)
#output:
[1, 3, 9, 27]
To further illustrate this explaination, one can consider the problem as the minimum number of weights to span the number space [0, 40]. It is evident that the number of things you can do with each weight is trinary /ternary (add weight, remove weight, put weight on other side). So if we write our (unknown) weights (A, B, C, D) in descending order, our moves can be summarised as:
ABCD: Ternary:
40: ++++ 0000
39: +++0 0001
38: +++- 0002
37: ++0+ 0010
36: ++00 0011
35: ++0- 0012
34: ++-+ 0020
33: ++-0 0021
32: ++-- 0022
31: +0++ 0100
etc.
I have put ternary counting from 0 to 9 alongside, to illustrate that we are effectively in a trinary number system (base 3). Our solution can always be written as:
3**0 + 3**1 +3**2 +...+ 3**N >= Weight
For the minimum N that this holds true. The minimum solution will ALWAYS be of this form.
Furthermore, we can easily solve the problem for large weights and find the minimum number of pieces to span the space:
A man drops a known weight W, it breaks into pieces. His new weights allow him to weigh any weight up to W. How many weights are there, and what are they?
#what if the dropped weight was a million Kg:
print find_weights(1000000)
#output:
[1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 202839]
Try using permutations for a large weight and unknown number of pieces!!
Here is a brute-force itertools solution:
import itertools as it
def merchant_puzzle(weight, pieces):
full = range(1, weight+1)
all_nums = set(full)
comb = [x for x in it.combinations(full, pieces) if sum(x)==weight]
funcs = (lambda x: 0, lambda x: x, lambda x: -x)
for c in comb:
sums = set()
for fmap in it.product(funcs, repeat=pieces):
s = sum(f(x) for x, f in zip(c, fmap))
if s > 0:
sums.add(s)
if sums == all_nums:
return c
>>> merchant_puzzle(40, 4)
(1, 3, 9, 27)
For an explanation of how it works, check out the answer Avaris gave, this is an implementation of the same algorithm.
You are close, very close :).
Since this is a puzzle you want to solve, I'll just give pointers. For this part:
Eg if (a,b,c,d) is the first set of values in comb, i need to check
a,b,c,d,a+b,a-b, .................a+b+c-d,a-b+c+d........ and so on.
Consider this: Each weight can be put to one scale, the other or neither. So for the case of a, this can be represented as [a, -a, 0]. Same with the other three. Now you need all possible pairings with these 3 possibilities for each weight (hint: itertools.product). Then, a possible measuring of a pairing (lets say: (a, -b, c, 0)) is merely the sum of these (a-b+c+0).
All that is left is just checking if you could 'measure' all the required weights. set might come handy here.
PS: As it was stated in the comments, for the general case, it might not be necessary that these divided weights should be distinct (for this problem it is). You might reconsider itertools.combinations.
I brute forced the hell out of the second part.
Do not click this if you don't want to see the answer. Obviously, if I was better at permutations, this would have required a lot less cut/paste search/replace:
http://pastebin.com/4y2bHCVr
I don't know Python syntax, but maybe you can decode this Scala code; start with the 2nd for-loop:
def setTo40 (a: Int, b: Int, c: Int, d: Int) = {
val vec = for (
fa <- List (0, 1, -1);
fb <- List (0, 1, -1);
fc <- List (0, 1, -1);
fd <- List (0, 1, -1);
prod = fa * a + fb * b + fc * c + fd * d;
if (prod > 0)
) yield (prod)
vec.toSet
}
for (a <- (1 to 9);
b <- (a to 14);
c <- (b to 20);
d = 40-(a+b+c)
if (d > 0)) {
if (setTo40 (a, b, c, d).size > 39)
println (a + " " + b + " " + c + " " + d)
}
With weights [2, 5, 15, 18] you can also measure all objects between 1 and 40kg, although some of them will need to be measured indirectly. For example, to measure an object weighting 39kg, you would first compare it with 40kg and the balance would pend to the 40kg side (because 39 < 40), but then if you remove the 2kg weight it would pend to the other side (because 39 > 38) and thus you can conclude the object weights 39kg.
More interestingly, with weights [2, 5, 15, 45] you can measure all objects up to 67kg.
If anyone doesn't want to import a library to import combos/perms, this will generate all possible 4-move strategies...
# generates permutations of repeated values
def permutationsWithRepeats(n, v):
perms = []
value = [0] * n
N = n - 1
i = n - 1
while i > -1:
perms.append(list(value))
if value[N] < v:
value[N] += 1
else:
while (i > -1) and (value[i] == v):
value[i] = 0
i -= 1
if i > -1:
value[i] += 1
i = N
return perms
# generates the all possible permutations of 4 ternary moves
def strategy():
move = ['-', '0', '+']
perms = permutationsWithRepeats(4, 2)
for i in range(len(perms)):
s = ''
for j in range(4):
s += move[perms[i][j]]
print s
# execute
strategy()
My solution as follows:
#!/usr/bin/env python3
weight = 40
parts = 4
part=[0] * parts
def test_solution(p, weight,show_result=False):
cv=[0,0,0,0]
for check_weight in range(1,weight+1):
sum_ok = False
for parts_used in range(2 ** parts):
for options in range(2 ** parts):
for pos in range(parts):
pos_neg = int('{0:0{1}b}'.format(options,parts)[pos]) * 2 - 1
use = int('{0:0{1}b}'.format(parts_used,parts)[pos])
cv[pos] = p[pos] * pos_neg * use
if sum(cv) == check_weight:
if show_result:
print("{} = sum of:{}".format(check_weight, cv))
sum_ok = True
break
if sum_ok:
continue
else:
return False
return True
for part[0] in range(1,weight-parts):
for part[1] in range(part[0]+1, weight - part[0]):
for part[2] in range( part[1] + 1 , weight - sum(part[0:2])):
part[3] = weight - sum(part[0:3])
if test_solution(part,weight):
print(part)
test_solution(part,weight,True)
exit()
It gives you all the solutions for the given weights
More dynamic than my previous answer, so it also works with other numbers. But breaking up into 5 peaces takes some time:
#!/usr/bin/env python3
weight = 121
nr_of_parts = 5
# weight = 40
# nr_of_parts = 4
weight = 13
nr_of_parts = 3
part=[0] * nr_of_parts
def test_solution(p, weight,show_result=False):
cv=[0] * nr_of_parts
for check_weight in range(1,weight+1):
sum_ok = False
for nr_of_parts_used in range(2 ** nr_of_parts):
for options in range(2 ** nr_of_parts):
for pos in range(nr_of_parts):
pos_neg = int('{0:0{1}b}'.format(options,nr_of_parts)[pos]) * 2 - 1
use = int('{0:0{1}b}'.format(nr_of_parts_used,nr_of_parts)[pos])
cv[pos] = p[pos] * pos_neg * use
if sum(cv) == check_weight:
if show_result:
print("{} = sum of:{}".format(check_weight, cv))
sum_ok = True
break
if sum_ok:
continue
else:
return False
return True
def set_parts(part,position, nr_of_parts, weight):
if position == 0:
part[position] = 1
part, valid = set_parts(part,position+1,nr_of_parts,weight)
return part, valid
if position == nr_of_parts - 1:
part[position] = weight - sum(part)
if part[position -1] >= part[position]:
return part, False
return part, True
part[position]=max(part[position-1]+1,part[position])
part, valid = set_parts(part, position + 1, nr_of_parts, weight)
if not valid:
part[position]=max(part[position-1]+1,part[position]+1)
part=part[0:position+1] + [0] * (nr_of_parts - position - 1)
part, valid = set_parts(part, position + 1, nr_of_parts, weight)
return part, valid
while True:
part, valid = set_parts(part, 0, nr_of_parts, weight)
if not valid:
print(part)
print ('No solution posible')
exit()
if test_solution(part,weight):
print(part,' ')
test_solution(part,weight,True)
exit()
else:
print(part,' ', end='\r')