If I deal some cards among four, three cards remain. If I deal among three, two remain and if I deal among five, two cards remain. How many cards are there?
The deck of playing card math puzzle
A number of cards has been lost from a complete pack. From the remaining cards, 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?
Recommended time to solve: 5 minutes.
Solution to the deck of playing card math puzzle
We don't proceed randomly and home in on the first key pattern that,
For both dealings of the cards among 3 people and 5 people, same number of 2 cards remain.
This means, when you divide the number of cards by either of 3 or 5, remainder in both instances is 2.
In other words,
The number of cards less 2 has both 3 and 5 as factors, that is, 3 x 5 = 15 as a factor.
Pattern 1: So the desired number of cards would be a multiple of 15 plus 2 and would be less than 52, which is the total number of cards in a deck.
We have taken care of two of the three given conditions.
The third condition states, the number when divided by 4 leaves a remainder of 3.
How is this fact related to a multiple of 5 plus 2?
Pattern 2: If you reduce a multiple of 5 by 1 and it becomes divisible by 4, then dividing this multiple of 5 plus 2 by 4 would give you a remainder of 3.
To make this clear let's take an example of multiple of 5 as 25. Reducing it by 1 would make it 24, divisible by 4. Dividing $25 + 2 = 27$ by 4 would give you a remainder of 3.
Apply this clue on the pattern 1 of the number and form the final pattern of the number,
Final pattern: The number would be a multiple of 15 plus 2 so that subtracting 1 from the multiple, result will be divisible by 4. Also the number will be less than 52.
A few trials
First multiple of 15 less 1, that is 14, is not divisible by 4.
Second multiple of 15 less 1, that is 29, is not divisible by 4.
But third multiple of 15 less 1, that is 44, is divisible by 4.
So our mystery number of cards is 44 + 3 = 47.
When it is divided by 3 or 5, on both occasions remainder is 2.
And when divided by 4, remainder is 3.
End note
The problem is simple, but the solution shows how the puzzle can be solved step by step by key pattern identification, mathematical reasoning and a few systematic trials.
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