The rarest Easy cells

An Easy board in Goaler's Tic-Tac-Toe crosses three big clubs with three big football nations. The core pool has 12 clubs and 9 nations, which makes 108 possible pairs, and four occasional clubs and three occasional nations take the total to 192. Most of those pairs have a crowd of answers. A few have almost none, and most of those involve England or Italy: English and Italian players have seldom left their home leagues, and Italian clubs could not sign foreign players from 1966 to 1980. This article counts every pair. It names no players, because these pairs are on real boards and finding the answers is the game.

How the count works

The counts use the rules Goaler's board builder uses. A player fits a pair when he made at least one senior appearance for the club, loans included, and is listed for the nation. He is a solid answer when he also has a senior cap for that nation, English Wikipedia's infobox confirms the link, and he did not join the club only this season. Uncapped nationals and caps for a former state such as Yugoslavia are accepted but not solid, and an Easy cell needs at least three solid answers. Our totals match the stored counts of the 3,075 club and country cells on Goaler's boards, except 49 Brazil cells stored before one player's Brazil caps were added to the data. They cover only the players in Goaler's data, men with pages on 20 or more Wikipedia language editions, so a few obscure answers are missing from every pair.

Nine of the 108 core pairs fall below that minimum, so they never appear on an Easy board even though both of their headers do: Manchester United with Germany or Italy, Arsenal with Argentina, Italy or Portugal, Juventus, Barcelona and Paris Saint-Germain with England, and Bayern with Argentina. An era rule keeps 24 more off. Every Easy cell needs at least two solid answers who played for the club in 2015 or later and are among the 1,000 best-known players in Goaler's fame rank; the table's last column counts only the first half of that. Barcelona and Italy, Real Madrid with Italy or the Netherlands, AC Milan and Germany, and Bayern and Italy have no solid answer at the club since 2015. Manchester City and Italy, Inter and England, and Liverpool and Portugal have one each, so they are blocked as well. Even a rich pair can fail: Inter and Brazil has 23 solid answers, eight of them since 2015, but only one of those eight is among the best known. Of the 15 pairs in the table, only Bayern and England can appear on an Easy board.

England and Italy at home

England is the clearest case: 611 players in Goaler's data are accepted under England, and each of the five English clubs in the core pool has between 65 and 79 English answers. The seven core clubs outside England have 33 between them, exactly half of Arsenal's 66: two at Paris Saint-Germain, three at Barcelona, four each at Juventus and Bayern, five at Inter, seven at Real Madrid and eight at AC Milan. At Juventus, only one of the four is solid.

Italian players stayed at home almost as firmly. The nine core clubs outside Italy have 49 Italian answers in all, while Juventus and Italy alone has 164. Manchester United, Arsenal, Real Madrid and Bayern have three each, and Chelsea, with 13, has the most. Spanish players travelled further: the ten core clubs outside Spain have 132 Spanish answers, 23 of them at Liverpool.

The other thin pairs have reasons of their own. Portugal is the smallest core nation, with 189 players in Goaler's data, and its 12 core-club pairs add up to 75 solid answers, the fewest of any core nation. Argentina has 30 solid answers at Inter, the most of any core pair that is not a home cell, but two each at Arsenal and Bayern. Manchester United and Germany has a single answer in Goaler's data, against seven or eight at Arsenal, Chelsea and Manchester City.

The occasional headers

The occasional headers, Atlético Madrid, Tottenham, Roma and Dortmund among the clubs and Belgium, Croatia and Uruguay among the nations, make the emptiest pairs of all. Four have no answer at all in Goaler's data: Bayern and Uruguay, Manchester City and Uruguay, Manchester United and Croatia, and Paris Saint-Germain and Croatia. Of the 84 pairs that involve an occasional header, 26 fall below the minimum.

Croatia, Belgium and Uruguay are small nation headers, with 179, 187 and 198 players in Goaler's data, and Dortmund, with 201, has the fewest of any club in the Easy pool. It has a single solid answer with each of France, Italy, the Netherlands and Uruguay. An Easy board holds at most one occasional header, and at most one Easy board in eight has one, so these pairs are rare twice over.

Where the answers pile up

At the other end are the home cells, where a club meets its own country. Juventus and Italy has 132 solid answers, Barcelona and Spain 118 and Real Madrid and Spain 117, and the thinnest home cell, Arsenal and England, still has 51. Away from home, the median core pair has 8 solid answers, and the four richest are Inter and Argentina (30), Barcelona and Brazil (29), AC Milan and Brazil (26) and Inter and Brazil (23).

The nation rules behind these counts matter most for players with caps for two teams, since a former state's caps never carry over, as capped by two countries explains. A solid answer under a nation needs only one senior cap; the men who collected the most are in the most-capped men.