We found this week’s puzzle in an old book and the author referred to it as “an old one of unknown origin”. If it was old and of unknown origin back then, we feel that it should be shared so that it’s not lost forever. And it’s an absolute beauty!
The problem is to solve the following cryptarithm or alphametic, where each letter represents a digit 0-9.
$$ EVE / DID = .TALKTALK... $$
Neither EVE nor DID has a leading zero, and EVE / DID has been reduced to its lowest terms. The result of the division is a recurring decimal with a repeating pattern of four digits.
There is one unique solution.
As usual you can post the answers as a comment on this website, reply to the post on Facebook, or retweet or reply on Twitter @quizmastershop.
Answer and explanation at 9.00 on Monday.
Thomas,
Well worked out, and if we’re honest, a slightly better explanation than the one we’re posting on Monday morning.
We had pointed out Rich’s error on the Facebook page where he’d also been commenting, rather than on here – perhaps we should.
Congratulations.
QMS team
Unfortunately, Rich’s answer above doesn’t meet the requirement that EVE/DID has already been reduced to its lowest terms, since 2 divides both 212 and 606.
What makes this problem significantly easier to solve by hand is recognizing that the decimal 0.TALKTALK… is equivalent to the fraction TALK/9999. (There is a similar fact for any repeating fraction. For example, 1/7=0.142857142857… repeats every 6 digits, so it is equivalent to 142857/999999.)
Therefore, TALK/9999 simplifies EVE/DID in lowest terms, so DID is a factor of 9999. The prime factorization of 9999 is 3*3*11*101; the only three-digit factors are 101, 303, and 909. From here, we consider three cases.
Case (1): Assume DID = 101.
Then TALK/9999 = EVE/101, so TALK=99 * EVE. However, this presents a contradiction. As Rich points out, 0.TALKTALKTALK is less than 1, so EVE must be less than DID. But E can’t be 0 [no leading 0s] and it can’t be 1 [since D=1 already] so E must be at least 2, and EVE must be at least 200, far greater than DID=101. So Case (1) is impossible.
Case (2): Assume DID = 909.
Then TALK/9999 = EVE/909, so TALK = 11 * EVE. Once again, we can find a new contradiction. The ones digit of TALK is K; the ones digit of 11*EVE must be E. (If this isn’t immediately clear, imagine writing out EVE + EVE0 in long-form addition.) Since letters can’t share the same value, E cannot equal K, and we have a contradiction. So Case (2) is impossible.
This leaves us with only the final case.
Case (3): Assume DID = 303.
Then TALK/9999 = EVE/303, so TALK = 33 * EVE. From here, we can deduce the remaining digits by considering the same things that were problems in the previous two cases.
As Rich points out, 0.TALKTALKTALK is less than 1, so EVE must be less than DID. But E can’t be 0 [no leading 0s] and it can’t be 3 [since D=3 already]. If E is at least 4, then EVE is at least 400, which is greater than DID=303. So E can’t be 4 or higher, which leaves only two possibilities: either E=1 or E=2.
If E is 1, then consider the ones digit of TALK = 33*EVE. Then K = 3. But this can’t be, since D=3 already. So E cannot be 1.
Therefore, we now have E=2 and K=6.
At this point, it’s easiest just to check each possible value for V (it must be one of {1,4,5,7,8,9} since {0,2,3,6} are already taken) and check the value of TALK in that case.
If V=1, then EVE=212 and TALK=33*212 = 6996, so T=K=6 and A=L=9, which can’t be; the numbers aren’t unique.
If V=4, then EVE=242 and TALK=33*242 = 7986.
If V=5, then EVE=252 and TALK=33*252 = 8316, so A=3=D, which can’t be; the numbers aren’t unique.
If V=7, then EVE=272 and TALK=33*272 = 8976, so L=7=V, which can’t be; the numbers aren’t unique.
If V=8, then EVE=282 and TALK=33*282 = 9306, so A=3=D, which can’t be; the numbers aren’t unique.
If V=9, then EVE=292 and TALK=33*292 = 9636, so T=9=V, and A=K=6, and L=3=D, which can’t be; the numbers aren’t unique.
It follows now that the unique solution to the problem is that 0123456789 translates to I_EDV_KTLA (with no 2 and no 5 used) and 242/303 = 0.798679867986…
Your answer doesn’t work because gcd(212,606) is not 1.
Ok, so there are a couple of things you can determine from the start:
To get a sub-1 floating point number, DID has to be greater than EVE. We also know that neither DID or EVE begin with a zero. This means that both D and E can only consist of the numbers 1-9, whilst I and V can be 0-9 (but not the same as D or E.
The DID/EVE combinations provide something like 4000 combinations ((9*10*9*10)/2).
At this point, I decided there was little point writing out/excluding values by hand, so (cheated?) by writing some python:
===
#!/usr/bin/python
from future import division
e=[1,2,3,4,5,6,7,8,9]
v=[1,2,3,4,5,6,7,8,9,0]
d=[1,2,3,4,5,6,7,8,9]
i=[1,2,3,4,5,6,7,8,9,0]
for te in e:
for tv in v:
for td in d:
for ti in i:
tx=(100*te)(10*tv)te
ty=(100*td)(10*ti)td
if(ty<=tx):
pass
else:
rx=“%.16f” % (tx/ty)
pm1=str(rx)[2:6]
pm2=str(rx)[6:10]
pm3=str(rx)[10:14]
pm4=str(rx)[14:18]
if(pm1pm2 and pm2pm3 and pm3==pm4):
py1=int(pm1[0:1])
py2=int(pm1[1:2])
py3=int(pm1[2:3])
py4=int(pm1[3:4])
if(py1py2 or py1py3 or py1py4 or py2py3 or py2py4 or py3py4):
pass
elif(py1te or py1tv or py1td or py1ti):
pass
elif(py2te or py2tv or py2td or py2ti):
pass
elif(py3te or py3tv or py3td or py3ti):
pass
elif(py4te or py4tv or py4td or py4ti):
pass
else:
print(“EVE / DID = .TALKTALKTALKTALK”)
print(“%s / %s = %s” % (tx,ty,rx))
===
This gives us:
EVE / DID = .TALKTALKTALKTALK
212 / 606 = 0.3498349834983498