Question

Can you answer the riddle

A small number of cards has been lost from a complete pack. If I deal among four people, three cards remain. If I deal among three people, two remain and if I deal among five people, two cards remain. How many cards are there?

Answer 1

the number of cards lost are 5 and the cards left are 47.

Explanation:

We are given that certain number of cards are lost.

When divided among the four people, 3 cards are remaining. This implies that the number of cards lost may be 1,5 or 9 then only we will get the 3 cards left. Going by this approach, the number of cards available should be 52–1=51; 52–5=47; 52–9=43

Case 1: when one card is lost (available cards will be 51)

When 51 cards are divided among four people will give 3 remainders (48+3) which satisfies the given condition.

When 51 cards are divided among three people will give no remainder. But we expect the remainder to be 2. So, this case cannot be true.

Case 2: When 5 cards are lost (available cards will be 47)

When 47 cards are divided among four people will give 3 as remainder (44+3) which satisfies the given condition.

When 47 cards are divided among three people will give 2 as remainder (45+2) which satisfies the given condition.

When 47 cards are divided among five people will give 2 as remainder (45+2) which satisfies the given condition.

Answer 2

1) The first filter to use is that if dealt between 5 people, there are 2 cards left over:

=> the number of cards must end in 2 or 7.

2) But if dealt between 4 people, 3 cards remain:

=> it is not possible for the number of cards to end in 2 as there cannot be a remainder of 3 with any even number, hence number must end in 7.

3) If the cards are dealt between 3 people to give n cards each, then 2 cards remain:

If 3n+remainder 2 ends in 7

=>3n ends in 5.

4) If 3n < 50 then the only possible numbers for 3n are 15 and 45.

We are told that a small number of cards are lost, so check 3n=45:

3n+ remainder 2= 47, and we are told that dealing between 4 leaves remainder 3:

47/4+= 11 remainder 3.

Hence the number of cards is 47.