Your second chance for freedom came with the Five Games of Zarball, in which you had to win at least one of three combinations. Again, you lost in a mistake up against the King.

However you won your freedom, and deserved it, in the intense Royal Zarball Tournament, where you made your way to victory.

Now you have taken the King's offer as the Supreme Dignifiably Appointed Royal Zarball Trainer of Excellence. and figured out the probability of hats being returned to the proper heads in The Royal Zarball Spectators crisis.

You are still the Supreme Dignifiably Appointed Royal Zarball Trainer of Excellence, and so the 1-on-1 game of speed, skill, strength and stamina is your job... now you are helping arrange the

**Village Zarball Tournament**- but so many people want to play in the tournament, and with each new player, it's less and less likely you are going to have a perfect power of two number of players. So you cannot have a perfectly even tournament, so you've set up a system of byes.

There will be tryouts in which a villager's basic skill at the game is determined and seeded, so that the tournament can be arranged fairly, and the players that have more skill than the others in their round are given a "bye" to the next round- in other words, they get to skip this round without playing anybody.

**(1 - Warmup Problem)**

179 villagers showed up for the zarball tournament... what is the least number of total matches needed to find 1 winner?

Hint: there is an easy way to solve this and a harder way... the riddle is finding the easy way, though the hard way works too ;D

Don't post solutions to problem #1 in your post, as the answers are right here, just check your answers ;D unless you have a totally different solution of course, or want to discuss the answers. Now onto a harder problem:

**(2)**

The way that the tournament worked, with "byes" based on skill (not always bying only 1 person if there was an odd number, sometimes 3, sometimes 5), it ended up with a Final Five. Your friend, Perry, is in the Final Five, and these are the Final Five and Perry's chances of beating each of them:

Perry

Sanders 1/2

Dave 3/4

Xavier 1/3

Ella 7/8

Assume that chances of beating someone are relative and stand when other people are facing each other. For example, Perry has a 3/4 chance to beat Dave and a 1/3 chance to beat Xavier. Thus Xavier is twice as good as Perry (1/3 = 1:2) and Perry is three times as good as Dave (3/4 = 3:1) so Xavier is 6 times better than Dave, thus Xavier has a 6:1 or 6/7 chance to beat Dave.

Remember, you are the Supreme Dignifiably Appointed Royal Zarball Trainer of Excellence, so it's up to you to set up the system of byes and brackets and such. Each round you may have to end up rearranging the entire brackets to suit the tournament right.

So, anyway, it's down to these 5 people. You can arrange it any way you like using brackets and byes.

How can you make it fair so that each person has an equal likelihood OR AS CLOSE AS POSSIBLE to winning?

**(3)**

Problem 3 is like Problem 2 in every way except the objective. Now you want to arrange the brackets to give your friend Perry the highest possible chance of winning. In this problem, there can only be two byes (for example: 5,

**bye 1**, 4/2=2, +1 = 3,

**bye 1**, 2/2=1, +1 = 2, 2/2 = 1 winner) and the same person cannot be byed twice.