Connect with us:

# Ponder This

Welcome to our monthly puzzles.
You are cordially invited to match wits with some of the best minds in IBM Research.

## December 2020 - Challenge

Let's consider a simplified version of the USA elector-based election system. In our system, the states are represented by a vector pop = [p1, p2, ..., pn] of odd numbers, given the number of eligible voters in any state.

Each state is allocated a given number of electors in the following manner: There are 1,001 electors in total, and they are assigned to states in proportion to their population. If p = p1+p2+...+pn, then state i receives (pi/p)*1001 electors, rounded down. The remaining electors are allocated one at a time for the states, ordered by the size of the remaining fractional elector (e.g., a state that got 3.5 electors will get another one before a state with 10.25 electors).

For example, if pop = [51, 61, 77, 89, 99] then the electors are [135, 162, 205, 236, 263].

There are two candidates in the election. If a candidate receives more than half the votes in a state, she gets all the electors for that state. The candidate with the most electors wins the election.

A vote is a vector [v1, v2,..,vn] that represents the number of votes a candidate received in all the states. The size of a vote is v1+v2+...+vn - the total number of votes for that candidate. A losing vote is a vote that results in losing the election (i.e., not getting enough electors). A losing vote is maximal with respect to a given population vector pop if any vote of a larger size is not a losing vote.

Your goal: Find a population vector p=[p1, p2, p3, p4, p5] on five states with 101<=pi<= 149 and a maximal losing vote v=[v1, v2, v3, v4, v5] such that the size of v is 71.781305% the size of p up to a millionth of a percent.

[p1, p2, p3, p4, p5]
[v1, v2, v3, v4, v5]

A bonus '*' for computing the maximal losing vote for the 2020 presidental election (where the populations are the eligble voters for each state, but the electors are computed by the non-simplified, real-world formula).

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: 30/11/2020 @ 12:00 PM EST
Solution: 04/01/2021 @ 12:00 PM EST
List Updated: 25/01/2021 @ 12:00 PM EST

Lorenz Reichel(30/11/2020 4:42 PM IDT)
Michal Politowski(30/11/2020 5:10 PM IDT)
Daniel Chong Jyh Tar(30/11/2020 5:29 PM IDT)
Guillaume Escamocher(30/11/2020 8:16 PM IDT)
*Dan Dima(30/11/2020 8:16 PM IDT)
Bert Dobbelaere(30/11/2020 8:18 PM IDT)
Uoti Urpala(30/11/2020 10:39 PM IDT)
Eden Saig(1/12/2020 2:31 AM IDT)
Colas Kerkhove(1/12/2020 4:22 AM IDT)
Xiao Liu(1/12/2020 8:20 AM IDT)
*Matt Rips(1/12/2020 8:33 AM IDT)
*Bertram Felgenhauer(1/12/2020 8:57 AM IDT)
Graham Hemsley(1/12/2020 3:15 PM IDT)
Benjamin Lui(1/12/2020 5:14 PM IDT)
S Wirth(1/12/2020 6:45 PM IDT)
*Jacob Siemons(1/12/2020 11:10 PM IDT)
Keith Schneider(2/12/2020 3:53 AM IDT)
Sean Egan(2/12/2020 5:24 AM IDT)
*Alper Halbutogullari(1/12/2020 1:25 PM IDT)
*andy greig(2/12/2020 2:24 PM IDT)
James Dow Allen(2/12/2020 2:56 PM IDT)
*David Greer(2/12/2020 3:52 PM IDT)
Paul Shaw(2/12/2020 4:06 PM IDT)
Hu Shuai(2/12/2020 4:26 PM IDT)
*Dieter Beckerle(2/12/2020 7:09 PM IDT)
Chuck Carroll(2/12/2020 11:26 PM IDT)
Willem Hendriks(3/12/2020 3:27 PM IDT)
Amos Guler(3/12/2020 4:30 PM IDT)
Yasodhar Patnaik(4/12/2020 4:09 AM IDT)
Liubing Yu(4/12/2020 9:01 AM IDT)
Turgay Yuktas(4/12/2020 11:22 PM IDT)
Karl D’Souza(4/12/2020 11:58 PM IDT)
Luke Pebody(5/12/2020 2:03 AM IDT)
Michael Branicky(5/12/2020 7:07 AM IDT)
Justin Gensler(5/12/2020 8:10 PM IDT)
Reiner Martin(6/12/2020 1:27 AM IDT)
*Latchezar Christov(6/12/2020 9:40 AM IDT)
*Sanandan Swaminathan(6/12/2020 10:19 AM IDT)
Cynthia Beauchemin(6/12/2020 7:54 PM IDT)
Rob Pratt(6/12/2020 8:07 PM IDT)
Gyorgy Gyomai(6/12/2020 9:14 PM IDT)
*Daniel Moolman(4/12/2020 1:18 PM IDT)
Victor Chang(7/12/2020 7:16 AM IDT)
Bryce Fore(7/12/2020 11:08 AM IDT)
Dipto Thakurta(8/12/2020 5:31 AM IDT)
Shouky Dan & Tamir Ganor(8/12/2020 8:26 AM IDT)
Ingo Ogertschnig(8/12/2020 11:41 AM IDT)
G.Samos(8/12/2020 12:49 PM IDT)
Alex Fleischer(8/12/2020 3:32 PM IDT)
Jerry Kearns(8/12/2020 5:54 PM IDT)
JJ Rabeyrin(8/12/2020 10:28 PM IDT)
Ilya Tarygin(9/12/2020 12:02 AM IDT)
Tyler Mullen(9/12/2020 1:24 PM IDT)
Almog Yair(9/12/2020 10:22 PM IDT)
*Eric Manalac(9/12/2020 12:08 PM IDT)
*Arthur Vause(10/12/2020 11:22 AM IDT)
Sri Mallikarjun J(11/12/2020 1:32 PM IDT)
Sebastian Bohm and Martí Bosch(11/12/2020 4:20 PM IDT)
Clive Tong(12/12/2020 11:22 AM IDT)
Walter Sebastian Gisler(12/12/2020 9:27 PM IDT)
*Motty Porat(13/12/2020 12:43 AM IDT)
Kamlesh Nahata(13/12/2020 1:53 AM IDT)
Hansraj Nahata(13/12/2020 7:48 PM IDT)
Thomas Pircher(13/12/2020 9:31 PM IDT)
Govind Jujare(14/12/2020 2:43 AM IDT)
*Li Li(15/12/2020 5:46 PM IDT)
Dan Ismailescu(16/12/2020 11:50 PM IDT)
Marco Bellocchi(17/12/2020 2:04 AM IDT)
Emma Barry(17/12/2020 2:48 PM IDT)
Daniel Bitin(19/12/2020 8:33 PM IDT)
Amir Sarid(20/12/2020 12:43 AM IDT)
*Florian Fischer(20/12/2020 3:20 AM IDT)
James Muir(20/12/2020 2:10 PM IDT)
Mathias Schenker(20/12/2020 5:21 PM IDT)
Nyles Heise(21/12/2020 6:50 AM IDT)
Chris Shannon(21/12/2020 11:01 AM IDT)
Harold Gutch(23/12/2020 7:51 PM IDT)
Shirish Chinchalkar(25/12/2020 11:50 PM IDT)
Paul Revenant(26/12/2020 9:05 PM IDT)
Karl Mahlburg(27/12/2020 3:41 AM IDT)
Fabio Michele Negroni(27/12/2020 5:29 PM IDT)
Harald Bögeholz(28/12/2020 4:28 AM IDT)
Sumanth Ravipati(28/12/2020 8:21 AM IDT)
Joaquim Carrapa(30/12/2020 3:55 AM IDT)
Franciraldo Cavalcante(30/12/2020 6:30 PM IDT)
Clement Heyd(31/12/2020 12:35 AM IDT)
Andreas Stiller(31/12/2020 1:20 AM IDT)
Zhou Hangbo(31/12/2020 3:33 PM IDT)
Li Wang(31/12/2020 3:33 PM IDT)
Kang Jin Cho(2/1/2021 3:06 AM IDT)
Oscar Volpatti(2/1/2021 7:17 AM IDT)
Vincent Beaud(3/1/2021 9:20 PM IDT)