## Archive for October 30th, 2009

### Problem 33 – Cancelling Fractions

October 30, 2009

49/98 = 4/8 if you cancel the 9s. This happens for few others fractions $n/d$ where $0, and we’ll also ignore $(n'\cdot 10)/(d'\cdot 10)$ for digits $n',d'$. Now take all these fractions that cancel like this, consider their product as a reduced fraction, and find the denominator. This is the challenge of problem 33.

Here’s what I came up with:

def gcd(a,b):
if b == 0:
return a
return gcd(b,a%b)

def solve():
fracs = [ ]
conv = lambda t:int("%s%s" % t)
for bothdig in xrange(1,10): # the digit that cancels
for numdig in xrange(1,10): # digit left in numerator
for dendig in xrange(1,10): # digit left in denom
nums = map(conv, [(numdig,bothdig),(bothdig,numdig)]) # consider bother orderings
dens = map(conv, [(dendig,bothdig),(bothdig,dendig)])
for v in [(n,d) for n in nums for d in dens if n<d]:
if v*dendig == v*numdig: # v/v = numdig/dendig
fracs += [ v ]
p = reduce(lambda l,r:(l*r,l*r),fracs) # multiply together solutions
return p / gcd(*p) # reduce the fraction, return denom


Certainly room for improvement, but it works.

### Problem 32 – Pandigital Products

October 30, 2009

We say that $n$ is a pandigital product if $n=a\cdot b$ and the string “abn” obtained by concatenating the base-10 representations of $a$, $b$, and $n$ contains all of the digits 1-9 exactly once (with no 0s). In problem 32 we are asked to find the sum of all pandigital products.

I ended up being not particularly fancy about this problem. Loop through enough values for $n$, and, for each $n$, loop through enough values for $a$, and see if you get a pandigital product with $n$, $a$, and $n/a$. We know that it is enough for $a$ to venture up to $\sqrt{n}$, but what about bounds on $n$? If $n$ has 5 digits, then to be a pandigital product, with factors $a$ and $b$, the factors must be either 1 and 3 digits, or 2 and 2 digits. But 9*999 and 99*99 aren’t 5 digit numbers, so we know we can stop looking for $n$ once we get to 10000. Similar reasoning shows that $n$ must have more than 3 digits, and so, in fact, to be a pandigital product, $n$ must have 4 digits (and since they must not be 0, and must be distinct, 1234 is the smallest possibility).

The following code seems to get the job done:

def ispandigprod(a,b,n):
""" assumes a*b==n """
# consider the digits in the string "abn"
# remove 0s and duplicates (with filter, set)
# if you've still got 9 elements, you're pandigital
return len(set(filter(lambda t:t, map(int, "%s%s%s" % (a,b,n))))) == 9

def solve():
""" return the sum of the pandigital products """
ret = 0
for n in xrange(1234,9876+1):
a = 1
included = False
while(a < n**.5 + 1 and not included):
if(n % a == 0):
b = n//a
if(ispandigprod(a,b,n)):
ret += n
included = True
a += 1
return ret


I’d honestly like a nicer solution for this problem. The loop I have is pretty stupid, and loops through far more choices than is worthwhile. In particular, I’d like to not think about any $n$ or $a$ that have a digit that is 0. To ignore $n$ with a 0 digit, one could replace the “included = False” line (apparently line 10) with

  included = str(n).count("0") > 0


This seems to cut the run-time down by about 20% (or more), so is maybe worthwhile. I still expect this code could be improved by a bit, but I think I’ll stop here for now.