You have all the clubs from a deck, 13 cards, and you can choose 2 from the deck and get paid their product, where all face cards are considered to be 0. you can pay $1 to reveal the difference of any two cards you choose, how much would you pay to play this game?

$11.

As the question states,

Number of clubs from the deck = 13

Number of selections which can be made = 2

Amount we are paid is the product of the cards value.

Value of Face Card = 0

Amount paid to reveal the difference of any two cards = $1

So,

The Maximum amount one person can get is when he get 9 and 10 as the value.

So,

Maximum Win = 9 x 10 = $90

For this,

We know we can guarantee the $90 total when we get the cards 9 and 10.

For this there are two cases:

Case 1 :

10 cards with some value and 3 cards with 0 value.

Starting with one card from the Face valued cards.

Here, we will be left with only 2 zeroes.

The maximum we can pay 10 + 1 = $11 to know that in maximum 11 steps we came to know the number 9 and 10.

Case 2 :

10 cards with some value and 3 cards with 0 value.

Starting with the cards other than Face Value cards.

Here, we will be left with 3 zeroes.

Therefore, in maximum 11 tries we can find out the required product of 90.

By paying $11.

Total Win $90 and Money Spent = $11.

Therefore we can get $90 by paying atleast $11.

560

hope this : )