## March 2022 - Challenge

** Primle **

In the popular game ** Wordle**, players attempt to guess a five-letter word. After each guess, the colors of the letter tiles change to reflect the guess. Green tiles represent a letter in the correct place; yellow tiles represent a letter present in the solution, but not in that place; and gray tiles represent a letter not present in the word at all. The submitted word has to be present in the game's dictionary; one cannot simply choose arbitrary letters.

We discuss a similar game, where instead of guessing a five-letter word, we attempt to guess an n-digit prime number (in the decimal system). As in the original Wordle, the guess **must** be a prime number. For this game, we want to pick a "good" initial guess. While there can be several ways to define "good" in this context, we choose the following: After the first guess, we obtain some pattern of green/yellow/gray connected to the digits in our guess. This pattern excludes some of the possible n-digit prime numbers. Let f(p, q) be the number of remaining possible solutions after the first guess, given that p was the guess and q is the solution.

For example, if the solution is the four-digit prime number 4733 and our initial guess was 3637, the resulting pattern will be yellow-gray-green-yellow, and the possible remaining solutions are 1733, 2731, 4733, 7039, 7331, 7333, 7433, 7933, 8731, 9733, 9739.

Note the following exclusions from the possible solution set of the example:

* 6733 is excluded since 6 was gray, and so 6 is guaranteed not to be part of the solution.

* 3733 is excluded since the first '3' was yellow, and so 3 is guaranteed not to be the first digit.

* 2239 is excluded since 7 was yellow, but is not present at all in 2239.

So in this case, f(3637, 4733)=11. An even better initial guess would have been 8731, which would have resulted in f(8731, 4733)=8.

For n-digits, we wish to find a prime number p which minimizes the expected value of f(p,q) for a uniformly generated prime q. That is, we wish to minimize E[p]\triangleq\sum_{q\in P}\frac{f(p,q)}{|P|} where P is the set of n-digit primes.

**Your goal**: Find this p for n=5 and compute E[p]. Return those two values as the solution.

**A bonus** "*" will be given for finding the solution for n=7.

We will post the names of those who submit a correct, original solution! If you don't want your name posted then please include such a statement in your submission!

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:**
28/02/2022 @ 14:00 PM EST

**Solution:**
04/04/2022 @ 12:00 PM EST

**List Updated:**
04/04/2022 @ 11:47 AM EST

#### People who answered correctly:

***Paul Revenant **(3/3/2022 1:55 PM IDT)

**Daniel Chong Jyh Tar **(3/3/2022 6:48 PM IDT)

***Alper Halbutogullari **(4/3/2022 1:02 AM IDT)

***Bertram Felgenhauer **(4/3/2022 2:48 AM IDT)

***Anders Kaseorg **(4/3/2022 3:11 AM IDT)

***Victor Chang **(4/3/2022 6:34 AM IDT)

***Bert Dobbelaere **(4/3/2022 9:21 AM IDT)

**Reiner Martin **(4/3/2022 10:08 AM IDT)

**Radu-Alexandru Todor **(4/3/2022 7:02 PM IDT)

**Jeremias Herrmann **(4/3/2022 9:14 PM IDT)

***Bertram Felgenhauer **(4/3/2022 9:56 PM IDT)

**Sanandan Swaminathan **(4/3/2022 11:26 PM IDT)

**Vladimir Volevich **(5/3/2022 2:17 AM IDT)

***Arthur Vause **(5/3/2022 10:24 AM IDT)

**Dan Dima **(5/3/2022 3:02 PM IDT)

**Vineet Dwivedi **(5/3/2022 3:55 PM IDT)

**Graham Hemsley **(5/3/2022 10:25 PM IDT)

**Michael Liepelt **(6/3/2022 12:15 PM IDT)

***Phil Proudman **(6/3/2022 2:44 PM IDT)

***Latchezar Christov **(6/3/2022 8:17 PM IDT)

***Amos Guler **(6/3/2022 8:59 PM IDT)

***Gary M. Gerken **(6/3/2022 9:17 PM IDT)

***John Goh **(7/3/2022 3:40 AM IDT)

***Dieter Beckerle **(7/3/2022 9:31 AM IDT)

***David Greer **(7/3/2022 1:16 PM IDT)

***Stéphane Higueret **(7/3/2022 3:55 PM IDT)

***Andreas Stiller **(7/3/2022 5:41 PM IDT)

***Chris Shannon **(7/3/2022 6:57 PM IDT)

**Ralf Jonas **(8/3/2022 8:14 AM IDT)

***Paul Shaw **(8/3/2022 3:35 PM IDT)

***Daniel Bitin **(8/3/2022 3:59 PM IDT)

**Vladimir Volevich **(8/3/2022 4:51 PM IDT)

**Clive Tong **(8/3/2022 6:50 PM IDT)

**Bruno Santos **(9/3/2022 2:09 AM IDT)

**Alexander Gramolin **(9/3/2022 6:36 AM IDT)

**Mathias Schenker **(9/3/2022 6:22 PM IDT)

***Peter Moser **(10/3/2022 10:09 PM IDT)

**Sachal Mahajan **(11/3/2022 1:12 AM IDT)

***Jan Fricke **(11/3/2022 10:31 AM IDT)

***Tim Walters **(12/3/2022 4:21 AM IDT)

***Nyles Heise **(13/3/2022 9:39 PM IDT)

**Bryce Fore **(13/3/2022 10:09 PM IDT)

***Liubing Yu **(15/3/2022 5:50 AM IDT)

***Sri Mallikarjun J **(15/3/2022 8:53 AM IDT)

**Markó Horváth **(15/3/2022 9:25 AM IDT)

***Alexander Marschoun **(17/3/2022 9:07 PM IDT)

***HC Zhu **(18/3/2022 1:03 AM IDT)

**Marco Bellocchi **(18/3/2022 9:58 AM IDT)

**Quentin Higueret **(18/3/2022 11:40 AM IDT)

**Palash Tyagi **(18/3/2022 9:06 PM IDT)

**Mark J. Murphy **(20/3/2022 2:56 AM IDT)

***Kai Guttmann **(20/3/2022 10:31 AM IDT)

***Dominik Reichl **(20/3/2022 6:14 PM IDT)

**Michael Branicky **(20/3/2022 10:32 PM IDT)

***Wolfgang Kais **(21/3/2022 9:02 PM IDT)

**Aviv Nisgav **(22/3/2022 8:54 AM IDT)

**Sanjay Rao **(22/3/2022 3:10 PM IDT)

**Stefan Wirth **(23/3/2022 1:20 PM IDT)

**Ananda Raidu **(24/3/2022 8:03 AM IDT)

**Fabio Filatrella **(24/3/2022 12:34 PM IDT)

**Kang Jin Cho **(24/3/2022 6:20 PM IDT)

**Greg Janée **(26/3/2022 9:14 AM IDT)

**Benjamin Lui **(27/3/2022 5:39 PM IDT)

**K S **(27/3/2022 7:36 PM IDT)

**Phong Kah Ho **(28/3/2022 4:38 AM IDT)

**Albert Stadler **(28/3/2022 5:22 PM IDT)

**Tamir Ganor & Shouky Dan **(28/3/2022 10:15 PM IDT)

***Motty Porat **(29/3/2022 3:33 AM IDT)

**Harald Bögeholz **(29/3/2022 5:21 AM IDT)

**Lorenz Reichel **(29/3/2022 6:39 PM IDT)

**Sebastian Bohm & Martí Bosch **(30/3/2022 10:55 AM IDT)

**Reda Kebbaj **(31/3/2022 9:11 PM IDT)

***Paolo Farinelli & David Dunkley **(31/3/2022 10:07 PM IDT)

**Todd Will **(2/4/2022 5:26 PM IDT)

***Jim Clare **(2/4/2022 11:01 PM IDT)

**Erik Wünstel **(3/4/2022 5:01 PM IDT)

**Franciraldo Cavalcante **(4/4/2022 5:20 PM IDT)

**Hansraj Nahata **(4/4/2022 7:27 PM IDT)