Here is a puzzle that I had a lot of fun composing. As it suggests, the YOB section of Part B is best done with the help of a computer.

I've provided some helps, but if you'd rather do the puzzle without them just ignore the three formulas in blue and the prime factorization aids directly below them.

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Joan's telephone number and my year of birth

Part A: Joan's telephone number

Joan said to Helen: "Will you ring me and let me know if you are able to come to the fête with me?"

"Gladly," said Helen, "but how will I remember your telephone number? I have no pencil with which to write it down, and I can seldom remember numbers."

"Well," said Joan, "maybe you can remember these four points about it, and work it out when you get home."

These are the four clues that Joan gave Helen:-

1. It consists of six digits, and no digit occurs more than once.

2. The product of the first three digits is equal to twice the product of the second three digits.

3. When added, the 1st, 3rd and 5th digits come to the same totals as do the 2nd, 4th and 6th.

4. The 1st digit is equal to the 3rd digit multiplied by the 6th.

What is Joan's telephone number?

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Part B: My YOB

This part is based on Part A and contains some interesting nonsense, as well as a serious component by which you can ascertain my year of birth ("YOB") – which was in the 20th century AD – and Part A's year of publication ("YOP") – which my YOB predates.

That serious component can be solved longhand…IF you have LOTS of time to kill, have LOADS of paper, a new pen, TONS of stickability, can concentrate for HOURS on end, can factorize and want to try to prove that man can beat machine – but solving with the help of a computer is highly recommended. So don’t complain if you take the longhand route and run out of midnight oil...you were warned!

I used Excel for my solution, but…

1. Excel lacks digit-manipulation functions for three of the puzzle's tasks, so I used these formulas that I found on the internet:-

(a) To display a number's reverse: =SUM(VALUE(MID(A1,ROW(INDIRECT("1:"&LEN(A1))),1))*10^(ROW(INDIRECT("1:"&LEN(A1)))-1))

(b) To display a number's digit sum: =SUM(VALUE(MID(A1,ROW($A$1:OFFSET($A$1,LEN(A1)-1,0)),1)))

(c) To display a number's digit product: =PRODUCT(VALUE(MID(A1,ROW($A$1:OFFSET($A$1,LEN(A1)-1,0)),1)))

They are array formulas. Type them in as usual, left-click the formula bar & press CTRL+SHIFT+ENTER: Excel then encloses them in braces {}.

Cell "A1" contains the number you wish to manipulate. Change it suit your spreadsheet…but don't change "$A$1".

2. Excel lacks a prime factorization function. To factorize multiple numbers concurrently I adapted Dr. Kardi Teknomo's "Prime Factors Using Spreadsheet" program, found at http://people.revoledu.com/kardi/tutorial/BasicMath/Prime/PrimeFactor.htm

Here are image links to scaled-down versions of my two Excel worksheets, showing only the factorization component (including formulas):-

"Factors": http://i52.photobucket.com/albums/g12/tomnrobn/JoanstelephonenumberandmyYOB-Factor.png

"Main": http://i52.photobucket.com/albums/g12/tomnrobn/JoanstelephonenumberandmyYOB-Main.png

These helps may not work in non-Excel spreadsheets – 1(a) & 2 fail in OOoCalc – in which case some manual input may be required.

Now to the puzzle…

Joan's telephone number, by some extraordinary 'coincidence', yields the same answer, "X", for the sum of the telephone number's three left-to-right paired digits and for the sum of the three right-to-left pairings! (eg, telephone number "278960" gives 27+89+60 l-to-r and 06+98+72 r-to-l). Furthermore, X holds special significance because when it is divided by the square that remains after deducting my YOB from Part A's YOP (and also when the reverse of that specially-significant number, X, is divided by the square's reverse), the result is a prime number that is a factor of Joan's telephone number, and its reverse!!

Had Helen known that, working out Joan's telephone number would have been a snap!

A little-known fact that would have flipped the doily under Helen's plate of afternoon tea gateaux is that of the amazing link between Joan's telephone number, the letters in Helen's name and the letters in Joan's name. If a name's letters are ascribed a value corresponding to their position in the alphabet (ie, a=1, b=2......z=26), the sum of those values becomes the name's "numeric value"; and if we multiply the numeric values of Joan's and Helen's names together and divide the result by the sum of the number of digits that appear in Joan's telephone number and those that do not (and don't forget the zero!), we arrive at that same specially-significant number – you guessed it – X, again!!

An even lesser-known fact that would have popped Helen's tea cosy right off the teapot is that the product of the values of the letters in Helen's name, divided by the product of the values of the letters in Joan's name, multiplied by the same prime factor of Joan's telephone number that I mentioned earlier, is – you guessed it again – that increasingly-specially-significant number, X, yet again!!

Even furthermore still (and knowing this would have enabled Helen to work out the number in her sleep!), the reverse of the difference between the product of the digits of my YOB and the product of the digits of Part A's YOP, multiplied by the reverse of the sum of the digits of the prime factors ("PFs") of the sum of my YOB and Part A's YOP, divided by the reverse of the product of the digits of the sum of the digits of the division of the difference between the number comprising the PFs (arranged in descending prime factor order) and its reverse by the sum of the digits of that difference, less a power of a root of the sum of the digits of the PFs, equals the sum of the digits not appearing in Joan's telephone number!!

What is my YOB and Part A's YOP? And what is the next solution each side of those years? All solutions are in the year range 0 to 3000 AD.

Solution check

The difference between the numeric value of Joan's name and the reverse of the division of the difference between the sum of the three YOB solutions and the sum of their digits by that "specially-significant number", X, equals the sum of the digits in Joan's telephone number.

Notes

1. Prime factors (PFs) include all repeat factors:-

eg, 1926 factorizes into 2x3^2x107: ie, 2x3x3x107…which, "arranged in descending prime factor order", becomes 107332.

2. Power and root exponents, and quotients, are positive integers.

3. The result of "the difference between" is an absolute value.

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Here is a link to a .pdf file of the above:

http://host-a.net/surfertje/JoanstelephonenumberandmyYOB-puzzle.pdf

Hope you'll enjoy it!

Cheers,

Tom

EDITS: (a) The first 13 relate to the original puzzle; (b) #14 = puzzle replaced with new version.