Katie had a pack of twenty four cards numbered 1- 24. She arranged the cards into 6 piles. No more than 2 piles had the same number of cards the numbers on the cards in each pile added to the same total. What was the total and how could this be solved

24 + 23 + 22 + 21 + . . . + 2 + 1 =

This is the sum of all the cards in the deck. This sum has to be evenly divided by the numbers of piles, so divide this number by 6, and that is your answer.

but what about the quantity of cards in each pile? figuring out that each pile adds up to 50 is the easy part!

To solve this problem, we need to find the total sum of the numbers on the cards and determine the number of cards in each pile. Let's follow these steps:

Step 1: Find the total sum of the numbers on the cards:
We know there are 24 cards numbered from 1 to 24. The sum of an arithmetic series can be found using the formula:

n * (a + l) / 2

Where n is the number of terms, a is the first term, and l is the last term.

In this case, we have n = 24 (the number of cards), a = 1 (the first card), and l = 24 (the last card). Plugging these values into the formula:

Sum = 24 * (1 + 24) / 2
Sum = 24 * 25 / 2
Sum = 600

The total sum of the numbers on the cards is 600.

Step 2: Determine the number of cards in each pile:
We are given that no more than 2 piles can have the same number of cards. To find the number of cards in each pile, we need to find divisors of 24 (excluding 1 and 24) since we have 6 piles.

The divisors of 24 are 2, 3, 4, 6, 8, and 12. We need to distribute these divisors among the 6 piles ensuring that no more than 2 piles have the same number of cards.

One possible arrangement could be:
Pile 1: 4 cards (divisor 2)
Pile 2: 4 cards (divisor 2)
Pile 3: 6 cards (divisor 3)
Pile 4: 6 cards (divisor 3)
Pile 5: 6 cards (divisor 3)
Pile 6: 4 cards (divisor 2)

In this arrangement, there are no more than 2 piles with the same number of cards, and the numbers on the cards in each pile also add up to the same total.

Therefore, the total is 600, and the cards can be arranged into 6 piles with the distribution mentioned above.