I have recently coded the 1st line of an aria from an opera composed by George Gershwyn and I am ready to share with you the following information:

1. The code is a simple substitution code, ten distinct letters were replaced by the digits 0,1,2…9.
2. One letter was left uncoded.
3. The line consists of six words.
4. When coded the 1st, 4th, and the 5th words become square numbers and do not use the non-coded letter ,
5. The other words are either multiples of a square number, bigger than one, or use the uncoded letter.

I dare you to find the name of the aria, the numbers in my coding (or alternate solution complying with the above terms) and to explain how it relates to puzzle's title.

(In reply to re(4): HINT by broll)

Post, part 2.

VIII. We are told that 'The other words are either multiples of a square number, bigger than one, or use the uncoded letter.' But all the digits of NOW occur in WOMAN and are therefore numeric. Hence NOW is a multiple of a square number, bigger than one, and if NOW fails this test, the candidate solution must be rejected.

IX. We also know that the letter S in BESS encodes as either 0 (in which case BE is a 2 digit square) or 4, in which case {B=0,E=1} or {B=3, E=8}. If WOMAN and MY exhaust both these digits then BESS cannot be constructed, and the candidate solution must again be rejected.

X. If MAN is known, the solutions for {WO} can readily be computed. They are as follows:

025: solutions{11,21,24,38,42,60,65,87,93}. After excluding repeated digits, {38,87,93} remain.
38025: (NOW=583=11*53, which is squarefree)
87025: (NOW=578=2*17^2, but M=0,W=8)
93025: (NOW=539=7^2*11, but M=0, O=3)

041: solution {32}
32041: (NOW=123=3*41, which is squarefree)

076: solutions{51,75}, 
51076: (NOW=615=41*3*5, which is squarefree)

081: solutions{58,67} but 58 repeats a digit.
67081: (NOW=176=2^4*11, but M=0 and A=8)

089: solutions{47,80} but 80 repeats a digit.
47089: (NOW=974=2*487, which is squarefree)

104: solution {23}
23104: (NOW=432=2^4*3^3)(but A=0, N=4)

204: solutions{39,91}
39204: (NOW=493=17*29, which is squarefree)
91204: (NOW=419=IsP)

209: solutions{41,88} but 88 repeats a digit.
41209: (NOW=914=2*457, which is squarefree)

249: solutions{37,94} but 94 repeats a digit.
37249: (NOW=973=7*139, which is squarefree)

264: solutions {43,85} but 43 repeats a digit.
85264: (NOW=458=2*229, which is squarefree)

276: solution{30}
30276: (NOW=603=3^2*67, but W=3 and O=0)

289: solutions{54,71}
54289: (NOW=945=3^3*5*7, but M=2, W=5)
71289: (NOW=917=7*131, which is squarefree)

384: solutions{16}
16384: (NOW=461=IsP)

401: solutions{40,89} but 40 repeats a digit.
89401: (NOW=2*3^2*11, but M=4,O=9)

416: solutions{38,92}
38416: (NOW=683=isP)
92416: (NOW=629=17*37, which is squarefree)

456: solution{13}
13456: (NOW=631=isP)

481: solutions{36,95}
36481: (NOW=163=isP)(W=9)
95481: (NOW=159=3*53, which is squarefree)

601: solutions{39,90} but 90 repeats a digit.
39601: (NOW=193=isP)

609: solutions{10,21} but 10 repeats a digit.
21609: (NOW=912=2^4*3*19, but A=0, and Y must be 4, so BESS cannot be constructed)

625: solutions{15,30,50,75} but all except 30 repeat a digit.
30625: (NOW=503=IsP)

681: solutions{43,84} but 84 repeats a digit.
43681: (NOW=134=2*67)

689: solutions{13,17}
13689: (NOW=931=7^2*19, Y=4 since M=6, S=0 since 4 has been used and we have 2,5,7 left, so BESS must be 2500.
But YOU=437=19*23, while IS=70=2*5*7, both of which are squarefree)
17689: (NOW=971=IsP)

824: solutions{53,71}
53824: (NOW=435=3*5*29)
71824: (NOW=417=3*139)

876: solution 15
15876: (NOW=651=3*7*31)

(NOTE: IsP denotes a prime number.)

So either I've somehow missed the vital combination, or the given problem lacks a solution, on my interpretation of its terms. 

 

Edited on July 14, 2012, 5:30 am

Posted by broll on 2012-07-14 04:44:02