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(3): HINT by Ady TZIDON)

Consider the squares WOMAN and MY.

A122986 in Sloane gives the squares mod 1000, i.e. all possible solutions to the digits MAN.

I.  We are told that MY is a 2-digit square. Hence we can cull all those members of A122986 which start with 5,7, or 9, since no 2-digit square starts with those numbers.
II. We can also cull all those having repeated digits since the digits of WOMAN are distinct.
III.Next, we can cull those which 'complete the square' within their digits, e.g. encode M as 1 and also contain a 6, since we need the second digit for the encoding of Y in MY, and it would not be available if already encoded as A or N.
IV. Now we have these remaining candidates: {
016 084 216 304 481 849
024 089 236 321 601 856
025 096 241 324 609 864
036 104 249 329 625 876
041 124 264 384 681 896
049 129 276 401 689 }
056 184 281 416 804 
064 201 284 436 809 
076 204 289 456 824 
081 209 296 476 836 
V.  We next eliminate those candidates which do not have corresponding squares in the range 10000 to 100000, i.e. {016 024 036 056 064 084 096 184 216 296 304 324 476 836} since they cannot encode MAN in WOMAN.
VI. Further we can delete all those candidates all of whose solutions which repeat a digit in the encoding of WOMAN e.g. 59049/66049, since the digits of WOMAN are distinct, i.e.:
{049 124 129 201 236 241 281 284 321 329 436 804 809 849 856 864 896}
VII. The remaining candidates are a more manageable 24, namely:{
025 204 384 609
041 209 401 625
076 249 416 681
081 264 456 689
089 276 481 824
104 289 601 876} that need to be considered in more detail. I will post further on these later.

Posted by broll on 2012-07-14 02:47:57