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In the spirit of lightweight prestidigitation, take a break to enjoy a gimmicky digital magic show.

 .'''''''''.---> _________________________              An ordinary 4-digit decimal
 : 3 9 1 8 :--->|                         |             number from the audience,
 '.........'--->|        Binarizer        |             3 9 1 8, is placed in a
           '--->|_________________________|             not-so-mysterious box.
                  | | | | | | | | | | | |
                .'''''''''''''''''''''''''.             12 equivalent digits emerge in a
                : 1 1 1 1 0 1 0 0 1 1 1 0 :             different number system (binary).
                '.........................'
                  | | | | | | | | | | | |               Into another box they go...
                 _V_V_V_V_V_V_V_V_V_V_V_V_
                |                         |----.'''''''.          ...only to reappear
                |         Presto!         |----: j k l :             as an equivalent
                |_________________________|----'.......'             3-digit number  j k l
                                               /   |   \             in another
                                              /    |    \            number system.
                                             / ____V____ \
                                            / |         | \          The middle digit  k
                                           (  | Changeo |  )         becomes an equivalent
                                            \ |_________| /          2-digit number  m n
                                             \    | |    /           in yet another
                                              \   | |   /            number system.
                                              .'''''''''.
                                              : j m n l :            The 4 digits  j m n l
                                              '. . . . .'            together form an
                                                | | | |              amount in yet another
                                            ____V_V_V_V____          number system.
                                           |               |
                                           |  Abracadabra  |         Now for the topper,
                                           |_______________|         after a magical moment
                                                   |                 in one last box, an
                                                   V                 equivalent single
                                                 .'''.               decimal digit  x
                                                 : x :               is presented to a
                                                 '...'               mesmerized audience

The digits 3, 9, 1, 8, j, k, l, m, n and x are all different and none are 0.


Just what do you suppose that mysterious decimal digit x might really be?


Shh, psst...
...it’s not really magic, it’s all done with numbers...
...the original audience number was later spotted in their seat unharmed...
...if this takes more than a few minutes to get solved, I surely will believe in magic.

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  • $\begingroup$ How can there be ten different digits if none of them are 0? $\endgroup$
    – somebody
    Oct 25, 2016 at 4:35
  • $\begingroup$ Good catch on an important clue, @somebody. (The digits are numerical nonetheless, not fingers for instance.) $\endgroup$
    – humn
    Oct 25, 2016 at 4:44

1 Answer 1

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Presto!

convert input to hex.
output (jkl) = f4e

Changeo!

convert middle digit of input to roman numerals.
"4" becomes "iv"; output (jmnl) = five

Abracadabra!

convert text to numeric.
"five" becomes "5"; output (x) = 5

3, 9, 1, 8, j, k, l, m, n and x are all different and none are 0.

3, 9, 1, 8, f, 4, e, i, v, and 5.

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3
  • $\begingroup$ +1 for your brilliant and clever solution. But, just for the discussion, can "Abracadabra" be considered a "numeric system" without doubt? (The "Changeo" one is similar but I think there is no problem there) $\endgroup$
    – marcoresk
    Oct 25, 2016 at 21:44
  • $\begingroup$ reading carefully, you see binarizer, presto, and changeo all require their input to be converted to an equivalent representation in a different numbering system. binarizer takes decimal digits, treats them as a complete decimal number, and outputs the binary equivalent. presto similarly converts binary input to the identical value in a different system. and changeo, while only operating on a single input digit, likewise converts that digit to its exact equivalent representation in yet another numeric system. $\endgroup$
    – Rubio ♦
    Oct 25, 2016 at 21:52
  • 1
    $\begingroup$ Abracadabra is described as taking its four input digits, treating them as an amount in a numeric system (that is to say, those four digits are a representation of a numeric quantity, which is what a numeric system is), and I think we can agree that it can unambiguously be converted to the final equivalent decimal digit in a well defined way, as the problem required. $\endgroup$
    – Rubio ♦
    Oct 25, 2016 at 21:55

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