Question

Twelve basketball players, whose uniforms are numbered 1 through 12, stand around the center ring on the court in an arbitrary arrangement. Show that some three consecutive players have the sum of their numbers at least 20.

Answer 1

Using the pigeonhole principle and the sum of the first 12 natural numbers, we can show that when the players' uniform numbers are summed up in groups of three, at least one group must have a sum of 20 or more because the average sum exceeds 19.5, which requires rounding up to an integer.

Step-by-step explanation:

To show that some three consecutive players have a sum of their uniform numbers at least 20, we can use the pigeonhole principle. Imagine the players are arranged in a circle around the center court. There are a total of 12 numbers, and the sum of these numbers is 1 + 2 + 3 + ... + 12, which is the sum of the first 12 natural numbers.

The formula for the sum of the first n natural numbers is n(n + 1)/2. Using this formula, the sum of the first 12 numbers is 12(12+1)/2 = 78.

If we divide the circle into four groups of three players, there are 4 groups. The average sum per group of three players will be the total sum divided by the number of groups, which is 78 / 4 = 19.5. Since the numbers are integers, the sum of the numbers in at least one group of three players will be at least 20 because you can't have half a number in this context, which implies a rounding up to an integer for at least one group.

This ensures that there must be at least one set of three consecutive players whose uniform numbers sum to 20 or more, thus fulfilling our requirement to show.

Answer 2

Shown

Step-by-step explanation:

Given that twelve basketball players, whose uniforms are numbered 1 through 12, stand around the center ring on the court in an arbitrary arrangement.

Let us consider consecutive numbers in this set.

`1+2+3 =6<20\\2+3+4 =9<20\\3+4 +5=12<20\\...\\\\5+6+7=18<20`

After this we find the totals are more than 20.

When 1 to 12 are arbitrarily arranged, there are chances that numbers from 6 and above are having consecutive numbers.

These totals are greater than 20

Hence shown that some three consecutive players have the sum of their numbers at least 20.

(i.e. starting from if we take)