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jackrider1598
14.11.2019 •
Mathematics
Find the number of ways of arranging the numbers
1,2, in a circle, so that the sum of any three adjacent numbers is divisible by 3. (two arrangements are considered the same if one arrangement can be rotated to obtain the other.)
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Ответ:
432
Step-by-step explanation:
Each number is a multiple of 3, or 1 more than a multiple of 3, or one less than a multiple of 3. These number we will call x's, y's, and z's.
If we add x + y + z, then we get x, because y and z will even each other out, since y is 1 more than multiple of 3 and z is one less.
Because of this, we can conclude that we have to arrange the numbers in a repeating pattern, like x, y, z, x, y, z, ... or z, y, x, z, y, x, ...
In this set 1, 2, 3, ..., 9, there are 3 x's, 3 y's, and 3 z's.
To fill up a pattern, there are 3 choices for the first x, y, z, 2 choices for second, then 1 choice.
3^3 * 2^3 * 1^3 = 27*8*1 = 216
This is equal for both patterns, so there are 216 * 2 = 432 ways to arrange the numbers.
Ответ:
First of all, note that all integers are either 0,1, or 2 modulo 3 (if you're not familiar with this terminology, it means that every integer is either a multiple of 3, or it is 1 or 2 away from a multiple of 3).
So, we can think of our numbers as
In order to make sure that the sum of any three adjacent numbers is divisible by 3, we have to make sure that any group of 3 three adjacent numbers contains a 0, a 1 and a 2. This is possible only if we arrange our 9 numbers in 3 groups of 3 numbers containing 0,1 and 2 exactly once, repeating always the same pattern.
For example, we could arrange our numbers following the pattern
or
We have
possible patterns. Suppose for example that we choose the pattern
One possible way of following this pattern would be the arrangement
In fact, we substituted every '0' with a multiple of 3 (3, 6 or 9), every '1' with a number 1 away from a multiple of 3 (1, 4 or 7) and every '2' with a number 2 away from a multiple of 3 (2, 5 or 8).
This means that, once we fix a patter, we have 3 choices for the first 3 slots, 2 choices for the next 3 slots, and the final slot will be fixed. So, we have
possible ways of following a fixed pattern. Since the number of patterns was 6, we have
possible arrangements.
Ответ:
triangle A is a right triangle
triangle B is a acute triangle
triangle C is a obtuse triangle
triangle D is a right triangle