IBM Research | Ponder This | November 2016 challenges

# Ponder This

## November 2016

<<October November December>>

Ponder This Challenge:

12 chefs (one for each month of the year) each has N sous chefs.
The chefs divide the sous chefs into three shifts: N1, N2 and N3 where N=N1+N2+N3 and no two numbers are the same.
After this partition, the three sets of sous chefs are randomly assigned to prepare the three daily meals (morning, lunch, and dinner).
For each meal, the chef who has more sous-chefs wins. If both chefs chose the same number - the result is a draw. When comparing two chefs, the one who won more meals wins.
For example: if N=9 and January's chef decided to split to 1,2,6 and February's chef decided to split to 2,3,4 then there are six possible permutations:

```Jan	Feb	meals winner	overall winner
1 2 6 vs 2 3 4	Feb, Feb, Jan	Feb
1 6 2 vs 2 3 4 	Feb, Jan, Feb	Feb
2 1 6 vs 2 3 4	Draw, Feb, Jan	Draw
2 6 1 vs 2 3 4	Draw, Jan, Feb	Draw
6 1 2 vs 2 3 4	Jan, Feb, Feb	Feb
6 2 1 vs 2 3 4  Jan, Feb, Feb   Feb
```

So the end result is that February wins. On the other hand, if the March chef chose 0,4,5, then he will win against February, but will lose to January.
IBM's Chef Watson analyzed the situation as a restaurant critic, and found that in every duel, the chef who wins is the one whose month appears first after a random date, i.e., when the June and July chefs are competing, June will win, but April will lose to December. In the event of a tie (like April and October), the culinary match ends with a tie as well.
Find the minimal N and a possible choice of the 12 chefs that can allow such a situation to occur.

Update (1/11): To clarify which month wins:

```    Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Jan  =   >   >   >   >   >   =   <   <   <   <   <
Feb  <   =   >   >   >   >   >   =   <   <   <   <
Mar  <   <   =   >   >   >   >   >   =   <   <   <
Apr  <   <   <   =   >   >   >   >   >   =   <   <
May  <   <   <   <   =   >   >   >   >   >   =   <
Jun  <   <   <   <   <   =   >   >   >   >   >   =
Jul  =   <   <   <   <   <   =   >   >   >   >   >
Aug  >   =   <   <   <   <   <   =   >   >   >   >
Sep  >   >   =   <   <   <   <   <   =   >   >   >
Oct  >   >   >   =   <   <   <   <   <   =   >   >
Nov  >   >   >   >   =   <   <   <   <   <   =   >
Dec  >   >   >   >   >   =   <   <   <   <   <   =
```

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We invite visitors to our website to submit an elegant solution. Send your submission to the ponder@il.ibm.com.

If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Challenge: 30/10/2016 @ 12:00 PM EST
Solution: 02/12/2016 @ 12:00 PM EST
List Updated: 07/12/2016 @ 12:00 PM EST

Robert Gerbicz (30/10/2016 07:25 PM IDT)
Francis Golding (31/10/2016 03:18 PM IDT)
Aviv Nisgav (31/10/2016 04:43 PM IDT)
Bert Dobbelaere (31/10/2016 05:36 PM IDT)
Ran Sharon (01/11/2016 12:27 PM IDT)
Leandro Araújo (01/11/2016 10:01 PM IDT)
Alex Fleischer (01/11/2016 10:33 PM IDT)
Yan-Wu He (02/11/2016 04:25 AM IDT)
Jesse Rearick (02/11/2016 07:08 AM IDT)
Shirish Chinchalkar (03/11/2016 12:04 AM IDT)
Andreas Stiller (04/11/2016 02:53 AM IDT)
David Greer (04/11/2016 07:07 PM IDT)
Harald Bögeholz (05/11/2016 08:45 PM IDT)
JJ Rabeyrin (05/11/2016 11:58 PM IDT)
Nis Jørgensen (06/11/2016 01:33 PM IDT)
Lorenz Reichel (06/11/2016 05:27 PM IDT)
Bryce Fore (07/11/2016 07:40 PM IDT)
Liubing Yu (09/11/2016 04:59 PM IDT)
Shouky Dan & Tamir Ganor (12/11/2016 02:36 PM IDT)
Luiz Friedrich (13/11/2016 01:34 PM IDT)
Itay Hazan (14/11/2016 07:04 AM IDT)
*Motty Porat (16/11/2016 01:25 AM IDT)
David F.H. Dunkley (17/11/2016 01:03 AM IDT)
Pål Hermunn Johansen (17/11/2016 04:56 PM IDT)
Thomas Satzger (18/11/2016 08:50 PM IDT)
Paolo Farinelli (20/11/2016 03:03 PM IDT)
Peter Gerritson (20/11/2016 08:33 PM IDT)
Daniel Bitin (21/11/2016 09:58 PM IDT)
Tyler Mullen (22/11/2016 06:51 AM IDT)
Teake Nutma (22/11/2016 09:22 PM IDT)
Stijn Vermeeren (23/11/2016 10:46 AM IDT)
Xu Zhiwei (24/11/2016 12:25 PM IDT)
Franciraldo Cavalcante (27/11/2016 03:56 AM IDT)
Wilder Boyden (27/11/2016 08:43 AM IDT)
John Spurgeon (27/11/2016 05:19 PM IDT)
Daniel Chong Jyh Tar (29/11/2016 05:11 AM IDT)
Christian Pape (29/11/2016 02:56 PM IDT)