# What is the smallest number of lies needed to solve a Dragon Joust puzzle?

I recently watched this video which presents a deduction "riddle" about dragon jousting. The rules for the puzzle (from the video) are as follows:

After centuries of war, the world’s kingdoms have come to an agreement. Every five years, teams representing the elves, goblins, and treefolk will compete in a grand tournament of dragon jousting. Every team will face each of the others once. The kingdom whose team wins the most matches will rule all of Center-Realm until the next tournament. [...]

You’ve been given the extremely important job of recording the scores for the first inaugural tournament. But the opening celebrations get a bit out of control, and when you wake up, you realize the games are already underway. Fortunately, no one has noticed your absence so far. However, you need to get up to speed quickly; if your boss, the head tournament official, finds out you’ve been sleeping on the job, you’ll lose your head.

After weighing your options, you decide to offer your life’s savings to [a wizard] in return for the information, giving him your blank scorecard to fill out. But before he can finish, your boss walks into the tent. You barely manage to hide the scorecard in time, and the wizard excuses himself.

Your boss chuckles. “Hope you didn’t believe anything Gorbak’s been saying— he’s been cursed to tell only lies, even in writing. Anyway, can you believe how low-scoring the tournament’s been? Every team has played at least once, yet not a single match with a combined score of more than [five points]! Anyhow, I’ll be back in a minute to review your scorecard.” You laugh along, and when he leaves you look at the partially completed card, now knowing every single number on it is wrong.

Along with the rules you get the partially filled scorecard with 6 columns and a row for each of the three teams.

The columns are:

• Number of games played

• Number of games won

• Number of games lost

• Number of games tied

• Total points scored across all games by the team of the row

• Total points scored across all games by opponents of the team of the row

The scorecard given in the video is

1 0 _ _ 6 _
2 _ _ 1 0 3
_ _ _ 0 0 1


However the top left square doesn't need to be filled out to solve the puzzle so the card

_ 0 _ _ 6 _
2 _ _ 1 0 3
_ _ _ 0 0 1


Would have been just as informative.

So this begs the question what is the fewest number of squares that Morbak could fill out in a scorecard such that the correct scorecard could be deduced? The solution to the puzzle does not have to be the same, it just has to be deducible from the start of the puzzle.

I have found a board with a unique solution using

$$8$$ lies

The board is

_ _ _ _ _ _
2 _ _ 0 0 1
2 _ _ 0 1 2

The solution to this board can be deduced as follows

Since each team must have played either one game or two games and team two and three have not played two game, both have played one game
_ _ _ _ _ _
1 _ _ _ _ _
1 _ _ _ _ _
And since each of these teams has tied at least one game, each of them has tied exactly one game with no wins or losses.
_ _ _ _ _ _
1 0 0 1 _ _
1 0 0 1 _ _
Now since team 1 played one game that tied the possible scores are:
0:0
1:1
2:2
(Greater scores are not possible without totaling to more than 5).
But since team three neither scored 1 nor had 2 scored against them the only possible score is 0:0
_ _ _ _ _ _
1 0 0 1 _ _
1 0 0 1 0 0
The same reasoning shows that team 2 tied their game 2:2
_ _ _ _ _ _
1 0 0 1 2 2
1 0 0 1 0 0
At this point it should be very clear that team two and team three have not played each other so the top row can be filled in to reflect the two matches team one must played.
2 0 0 2 2 2
1 0 0 1 2 2
1 0 0 1 0 0

Here is my strategy for coming up with this board.

If you fiddle around with the board a bit you will notice that the games section (the first four columns) is pretty easy to fill in with only a few clues. (3 clues (0 1 1 in the ties column) is enough to fill the entire section). And the score section (the last three columns) tends to be the hard part.
Additionally ties seem to give us the most information in the score box since knowing a game was tied restricts it to 3 outcomes. So it seems like a good idea to have as many games tie as possible.
Our options for all ties are pretty limited, either each team plays two games, or two teams play one game and one two. The best I could do with three ties is
_ _ _ 0 3 4
_ _ _ 1 1 2
_ _ _ 1 1 2
Which is very light on the game section, but fills the entire score section.
With two ties I got the answer with $$8$$ lies presented above.

• Seems like 8 lies is the least amount I can come up with too. Just wanted to point out that in your 3 tie example, the treefolk and goblins are also playing against each other, so the elves can get a score of 2 and 2 (1:1 ties with treefok and goblins) and the treefolk and goblins can each get a score of 3 and 3 (1:1 tie with elves and 2:2 tie with each other) in addition to the 0 0 scenario I'm assuming you discovered so you'll need more than 9 clues for that one. Feb 5, 2020 at 0:36