Puzzle
7 minute read

Ponder This Challenge - February 2019 - Debater's score

One of the databases project DEBATER's datasets has been opened to the public: the Wikipedia Oriented Relatedness Dataset (WORD), which contains 19,276 pairs of concepts and their relatedness.

The relatedness score in the dataset has only 64 different values. (See a screenshot from the dataset below.)

Assuming this is a result of the fact that only 14 people were used to label the data, one would expect to get 65 different scores. How many people are needed to make the number of possible scores a value that contains all 10 digits, with each digit being used exactly once?

Solution

  • There are 15 possible scores for 14 viewers: 0/14, 1/14, ..., 14/14; but there are more, like 4/13, when some viewers don't vote.

    In some cases, we get exactly the same score from different numbers of active viewers, like 3/7 = 6/14.

    The number of different scores is Phi(n), which is the sum of the Euler's totient function phi(n). For details, read more about the Farey Sequence.

    The smallest solution (barring leading zero) is 59978 viewers, which gives 1093486725 different scores.

    Deron Stewart suggested to generalize it to hexadecimal, where the first answer is n=1956027711, which gives 1023BA5E9687DC4F (in base 16) different scores.

    An interesting answer is 5927659249 viewers, which gives 9438715260CAEFBD (in base 16), where the letters (A-F) and the numerals (0-9) are grouped.

    Oscar Volpatti sent us the following table which shows all extremal solutions for bases up to B = 20:

    |----|----------------|----------------------|
|  B |  n_min, n_max  |  PHI(n)+1  (base-B)  |
|----|----------------|----------------------|
|  2 |              1 |                   10 |
|----|----------------|----------------------|
|  3 |              5 |                  102 |
|    |              7 |                  201 |
|----|----------------|----------------------|
|  4 |             21 |                 2031 |
|    |             25 |                 3021 |
|----|----------------|----------------------|
|  5 |              - |                    - |
|----|----------------|----------------------|
|  6 |            186 |               120453 |
|    |            367 |               514203 |
|----|----------------|----------------------|
|  7 |            656 |              1053624 |
|    |           1473 |              5416320 |
|----|----------------|----------------------|
|  8 |           2686 |             10273465 |
|    |           7284 |             75416023 |
|----|----------------|----------------------|
|  9 |              - |                    - |
|----|----------------|----------------------|
| 10 |          59978 |           1093486725 |
|    |         175048 |           9314028765 |
|----|----------------|----------------------|
| 11 |         298075 |          104A9756832 |
|    |         962264 |          A9401653278 |
|----|----------------|----------------------|
| 12 |        1575794 |         10234658A7B9 |
|    |        5356135 |         B8A039246571 |
|----|----------------|----------------------|
| 13 |              - |                    - |
|----|----------------|----------------------|
| 14 |       51562679 |       1037C9DA62B845 |
|    |      190486373 |       DC78296304A51B |
|----|----------------|----------------------|
| 15 |      311598486 |      1026EDBC957483A |
|    |     1197160093 |      EDC934BA2516870 |
|----|----------------|----------------------|
| 16 |     1956027711 |     1023BA5E9687DC4F |
|    |     7770855673 |     FEBAC801D5962473 |
|----|----------------|----------------------|
| 17 |              - |                    - |
|----|----------------|----------------------|
| 18 |    85091548389 |   1023G674E5CH8F9ABD |
|    |   359156254528 |   HGFC081746EB9A5D23 |
|----|----------------|----------------------|
| 19 |   587048125686 |  10234EDAC85H9GB6I7F |
|    |  2547280905034 |  IHGFA9CB2ED47635108 |
|----|----------------|----------------------|
| 20 |  4164319889998 | 1023469JF8BAIE7HGCD5 |
|    | 18547498826013 | JIHFDE159C8G04B63A27 |
|----|----------------|----------------------|
| 21 |              - |                    - |
|----|----------------|----------------------|

Solvers

  • Dan Dima (29/01/19 05:43 PM IDT)
  • *Uoti Urpala (29/01/19 05:43 PM IDT)
  • Reiner Martin (30/01/19 01:00 AM IDT)
  • *Anders Kaseorg (30/01/19 01:29 AM IDT)
  • *Alper Halbutogullari (30/01/19 02:32 AM IDT)
  • Deane Stewart (30/01/19 04:48 AM IDT)
  • *Deron Stewart (30/01/19 05:42 AM IDT)
  • *Vladimir Volevich (30/01/19 11:57 AM IDT)
  • *Thomas Huet (30/01/19 02:49 PM IDT)
  • *Hugo Pfoertner (30/01/19 02:51 PM IDT)
  • Louis Faucon (30/01/19 04:16 PM IDT)
  • Alexander Daryin (30/01/19 04:59 PM IDT)
  • Emma Kee (30/01/19 06:27 PM IDT)
  • Eden Saig (30/01/19 09:56 PM IDT)
  • *Harald Bögeholz (31/01/19 01:46 AM IDT)
  • Antoine Bodin (31/01/19 06:57 PM IDT)
  • Wolfgang Kais (31/01/19 07:30 PM IDT)
  • *Motty Porat (01/02/19 03:34 AM IDT)
  • *Thomas Egense (01/02/19 05:18 PM IDT)
  • Kurt Foster (01/02/19 07:18 PM IDT)
  • Luke Pebody (02/02/19 12:26 AM IDT)
  • *Michael J Swart (02/02/19 01:01 AM IDT)
  • Rashid Naimi (02/02/19 05:39 AM IDT)
  • *Jess Thrysoee (02/02/19 02:57 PM IDT)
  • Victor Chang (03/02/19 02:49 AM IDT)
  • *David F.H. Dunkley (03/02/19 03:09 AM IDT)
  • *Haye Chan (03/02/19 05:11 AM IDT)
  • Hyung Sik Hwang (03/02/19 05:52 AM IDT)
  • *Dominik Reichl (03/02/19 12:57 PM IDT)
  • Kipp Johnson (03/02/19 05:42 PM IDT)
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  • Joseph DeVincentis (06/02/19 06:29 AM IDT)
  • *Boris Silantyev (06/02/19 07:57 AM IDT)
  • *Shouky Dan & Tamir Ganor (06/02/19 08:13 AM IDT)
  • Rogerio Ponce da Silva (06/02/19 11:52 AM IDT)
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  • JJ Rabeyrin (06/02/19 11:10 PM IDT)
  • *Peter Gerritson (07/02/19 10:36 PM IDT)
  • Nyles Heise (07/02/19 11:04 PM IDT)
  • *Kang Jin Cho (08/02/19 08:09 AM IDT)
  • Erik Wünstel (08/02/19 11:57 AM IDT)
  • Steven Langerwerf (09/02/19 10:03 PM IDT)
  • Susan Brommer (10/02/19 12:03 PM IDT)
  • *Jingran Lin (10/02/19 12:57 PM IDT)
  • Eric Edward Pauley (10/02/19 01:36 PM IDT)
  • *Sanandan Swaminathan ((12/02/19 10:45 AM IDT)
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  • *Bert Dobbelaere (12/02/19 11:34 PM IDT)
  • Roger Duthie (13/02/19 01:55 PM IDT)
  • Siddhartha Srivastava (13/02/19 06:46 PM IDT)
  • *Tom Sirgedas (14/02/19 01:21 AM IDT)
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  • Armin Krauß (14/02/19 03:27 PM IDT)
  • Mathias Schenker (14/02/19 03:27 PM IDT)
  • *Marco Bellocchi (14/02/19 09:03 PM IDT)
  • Kenneth Kwok (15/02/19 08:39 AM IDT)
  • Fabian Thut (15/02/19 11:04 PM IDT)
  • *Jeff Boyle & Todd Will (16/02/19 03:07 AM IDT)
  • *Mykola Zubach (16/02/19 08:53 PM IDT)
  • *Tom Chen (18/02/19 10:44 AM IDT)
  • Clive Tong (18/02/19 10:35 PM IDT)
  • Greg Janée (18/02/19 10:57 PM IDT)
  • Almog Yair (19/02/19 06:10 PM IDT)
  • *David Greer (20/02/19 01:35 AM IDT)
  • Andreas Knüpfer (20/02/19 10:31 PM IDT)
  • Manoj Kataria (20/02/19 10:46 PM IDT)
  • Reda Kebbaj (21/02/19 08:46 PM IDT)
  • Mike Chirozzi (21/02/19 10:58 PM IDT)
  • Lachezar Hristov (22/02/19 05:10 PM IDT)
  • Jack Baumruk (25/02/19 06:25 AM IDT)
  • Anunay Kulshrestha (25/02/19 10:59 PM IDT)
  • *Vladimir M. Milovanović (26/02/19 12:37 AM IDT)
  • Rickard Björklund (28/02/19 12:56 AM IDT)
  • *Oscar Volpatti (28/02/19 03:19 PM IDT)
  • Radu-Alexandru Todor (01/03/19 03:37 AM IDT)
  • *Li Li (01/03/19 03:12 PM IDT)

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