# Math Question: How to Guarantee a win in an NFL Football Pool

My friends and i are in a football pool where each week you have to pick a loser in a game. At the beginning of the year you purchase how many picks you want - the number of picks you buy determines how many choices you get each week to choose a loser. If you get it right your pick is still alive and you move to the next week. Get it wrong (i.e the team you chose won instead of lost), and your pick is out. We are trying to determine how many picks you would have to buy to garaurantee that you'd win. There are  13 teams in the NFL and there's 17 weeks of play.
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Commented:
I'm not sure I understand all of the rules.

What do you mean "your pick is out"

Does that mean the team that I picked is no longer a choice I can make from the pool for the remainder of the season?
Or
Does that mean I have one less choice each week for the remainder of the season?
Or
Does it mean something else?

Also "number of picks you buy determines how many choices" - I assume one pick=one choice?

Also, 13 teams?  Since that's not all of them. Do you only pick between teams in the subset of games that are between those 13, or between all games or between games where they don't play each other?

If you can pick from games they play each other, are you allowed to pick both sides  (assuming you have 2 picks left?)
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Commented:
Can you pick both teams in a particular game to lose.
Can you place multiple picks (like 2^16 = 65536) on a single game.

If so, a possible strategy would be to put half you picks on each side of any game.
You will keep (and lose) half of your picks every week.
This will guarantee you have one pick left at the end of the season.
This is probably not an exciting or efficient strategy.

Otherwise, this is a fair and difficult contest.
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Author Commented:
Thanks for the quick response:

- Your "pick" is your choice each week. So at the beginning of the year i "buy" a "pick" which means that i now have a choice on a game - i have to choose a loser and if i get it right, i advance to the next week. I can put choose both sides if i want, yes.
- Sorry about the 13 there, not sure what i was thinking. You can choose between the games of all 32 NFL teams.

Let me know if i didn't answer all of the questions there.
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Author Commented:
You can pick both teams in a game to lose - therefore gauranteeing that you get at least one pick through. You can also pick multiple teams.
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Commented:
are you required to use all of your picks every week?

If not, you could buy 18 picks.  Each week,pick any one game and bet on both sides.

You'll lose one pick each week but remain with one left over at the end.
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Commented:
This website may help:   http://www.survivorgrid.com/

Apparently this is called a Survivor Pool.
It seems to be a true balance of luck and skill.
It is a fair contest assuming honest management.

The only honest way to assure a win would be to buy a massive number of picks.
If you do that, you would likely wind up splitting you own money with one or more lucky and/or skillful players.

The way to win dishonestly is to pick (or change your pick) after the game is over.
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Commented:
>>  If not, you could buy 18 picks.  Each week,pick any one game and bet on both sides.

You would lose 9 picks the first week.
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Author Commented:
Yeah this is what i'm trying to determine - how many picks would i need to buy gauranteeing a win at the end of the season, assuming that each week i'm going to lose half of my picks.
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Commented:
That was in my fist post:  2^16 = 65536
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Commented:
>>> You would lose 9 picks the first week.

That's why I asked "are you required to use all of your picks every week?"

if you aren't required to use all of your picks every week then your picks go like this...

You use 2 and save the other 16, You lose 1, keep1.

Next week, use 2, save 15, lose 1, keep 1
Next week, use 2, save 14, lose 1, keep 1
etc...
Last week, use 2, save 0, lose 1,  finish with 1 pick left for the win.

Again, not very fun - same idea as buying the exponential when you DO have to use them all.
But, if there is a "save" loophole, it's cheaper.
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Author Commented:
Yes you have to use all of your picks each week
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Author Commented:
How did you arrive at that number d-glitch?
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Commented:
>>> yes you have to use all of your picks each week

then d-glitch's  2^(N-1) is correct  for N weeks
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Author Commented:
The other important factor is that 6 teams get a bye each week, so there's only 26 teams playing 13 games.
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Commented:
actually I think you would need 2^N not 2^(N-1) to make sure you finish the Nth week with 1 pick left

So  2^17= 131072

If you go with 2^16, you'll go into the 17th week with just 1 pick left so you'd have to guess at the last one and maybe not win.

The number of teams playing isn't actually important at all (as long as there is at least one game), you are putting all of your picks on just one game, but split to both sides.
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Commented:
>>  6 teams get a bye each week

That's not quite true, and it doesn't matter.
There are no byes the first and last few weeks of the season.
You still need to allow for losing half you picks every week, so you need to start with 2^16.

A more important question is:  What happens in the case of a tie?
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Author Commented:
i tie is a loss from my understanding.
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Commented:
d-glitch are you sure about the 2^16? i.e 2^(N-1)

Just simplifying:  let's say I have a season consisting of just one game.

That would mean I would start with 2^0 =1 pick so I can't pick both sides to guarantee a win

I think you need 2^N instead to get the guarantee all the way through
So 2^17 = 131072
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Commented:
>>> i tie is a loss from my understanding.

"Loss" in what sense?

A loss for the team, meaning a win for you?
Or a loss for the pick, meaning a loss for you?

If a ties mean a loss of the pick, then there are no guarantees.
It's exceedingly unlikely but possible for every game to end in a tie, all picks would be lost - meaning no person could win the pool
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Commented:
I may have clarified the answer but d-glitch should at least get a split

However, with the ambiguity around ties, I'm not sure we have an answer yet anyway.
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Commented:
>>   So 2^17 = 131072

I agree.  Off by 1 (in the exponent).
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Author Commented:
Sorry, a little quick on the draw to close the question.  I'm not 100% sure about ties., but i think you're right there - if the games all tie, then everyone would be out. YOu have to pick the losers of the games.......if you get that right, your pick, or entry, is safe and you progress to next week. Need to know number of picks to buy to gaurantee winning at week 17. I think the 2^17 is the correct answer.
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