Hot Dog Contest

There is a solution based on just the first and last statements!

The first statement tells us that Bill did not come last.

The third statement tells us that Joey ate more hot dogs than Bill.

And so Joey must be first.

David did not eat the highest. So, Bill and Joey are in the race.but Joey eats more than Bill. So, Joey is the winner.

Let us provide ranks $1,2$ and $3$ and check the possibilities one by one

Bill did not eat the fewest hot dogs.

• This means that Bill can have a rank : $\{1,2\}$

David did not eat the most number of hot dogs.

• So David's rank can be : $\{2,3\}$

Joey ate more hot dogs than Bill.

• Thus, Joey's rank numerically is one less than Bill's rank and can be : $\{1-1,2-1\} = \{0,1\}$

But there's no such a rank as a $0$ rank since we have assumed the ranks to be $1,2$ and $3$ only. Thus we conclude that Joey's rank is $1$ , Bill's rank is $2$ and David's rank is $3$ . So Joey ate the most number of hot dogs and won the contest.

Bon Appétit !

let

$\text {rank 1= fewest number of hotdogs}$

$\text{rank 2 = more than fewest but less than most number of hotdogs}$

$\text{rank 3 = most number of hotdogs}$

From the first statement: Bill maybe rank $2$ or $3$ .

From the second statement: David maybe rank $1$ or $2$

From the third statement:Joey must be in rank $3$ because Bill is maybe in rank $2$ or $3$ , and we know that rank $3$ is the highest. This means that Bill is rank $2$ and David is rank $1$ .

• the first line states that bill was not the third finalist of the hot dog contest it means that he would be second or first
• the second line states that david was not the first finalist means he would be third or second
• the third line states that joey ate more than bill.
• bill had the possibility of first or second rank and then now when we know that joey is more than bill means bill holds the second rank and joey the first rank

Or you can just look at the second and last statements! David did not eat the most number of hot dogs and neither did Bill, considering that Bill ate fewer hot dogs than Joey according to the last statement, so Joey is the only possible person who ate the most number of hot dogs.

Bill did not eat the fewest hot dogs and Joey ate more than him means Joey ate most number of hot dogs.

Bill did not eat the fewest hot dogs but Joey ate more than Bill.

It is known that David did not eat the greatest number of hotdogs, thus we can conclude he is not among the candidates for winners.

The winners could be Bill or Joey.

It is know that Joey ate more than Bill, thus we can conclude that he ate more than both Bill and David.

Thus, the winner is Joey.

"David did not eat the greatest number of hot dogs", so Dave is out.

"Joey ate more hot dogs than Bill", so Bill is out.

If Bill did not eat the most and Dave did not eat the most, then Joey must be the one who ate the most