First, a little background
"Stryktipset" is a popular form of football (soccer) gambling in Sweden, but I'm sure similar games exist in many other countries. The concept is simple: out of a list of thirteen games, the player needs to pick the correct winner, or whether the game ends in a draw. The notation for this is 1 for home team win, X for a draw, and 2 for an away win.
The player competes against the player pool, where set percentages of the total prize pool gets divided among those who get 10, 11, 12 and 13 corretly picked games.
A common way to play is through "guarantees" and "semi-guarantees", meaning that for every game, the player can choose to play all three alternatives, or at least two of them.
So for instance, playing game 1 as 1X2, game 2 as X2 and the rest as "single choice", yields 3*2*(1^11) combinations, for 6 ways to get all 13 right, or 6 "rows", and thus a cost of 6 times the base cost. This is called an M, or Mathematical, system, whereas just picking one option per game is a "single".
To the point
Now, a common puzzle is "what is the lowest amount of rows that guarantees the player 5 correct games?" Many responders go by their gut feel of "guaranteeing" five games and then letting the remaining 8 be arbitrary, for 5^3 = 243 rows. But, if one plays just three singles that don't overlap, this turns out to be enough to guarantee at least 5 correct games. This can easily be seen by imagining one single with all 1's, one with all X's and one with all 2's. Naturally, at least one of the signs needs to show up at least five times.
But what I wonder is this:
What is the lowest amount of rows that guarantees 10 correct answers?