Write a number 1,3 5,7,9,11,13,15 or 17 in each circle so that of all circles in each side of triangle is 30
Answers
Step-by-step explanation:
The sum of all nine given numbers is 81.
The sum of all three sides of the triangle is 3 x 30 = 90
Three numbers in the corners will be counted twice in the sum of 90,
so the sum of the three numbers in the corners must be 90 - 81 = 9
The only way to get a sum of 9 with three of the given numbers is to take 1, 3 and 5. So those are the three numbers in the corners. It doesn't matter which of these three numbers gets put in each corner.
Consider the side with 1 and 3 at its ends: the sum of the remaining two circles must be 30 - 1 - 3 = 26.
The only way to get 26 with the remaining numbers is to use 17 and 9, or 11 and 15.
Consider the side with 3 and 5 at its ends: the sum of the remaining two circles must be 30 - 3 - 5 = 22.
The only way to get 22 with the remaining numbers is to use 7 and 15, or 9 and 13.
Consider the side with 1 and 5 at its ends: the sum of the remaining two circles must be 30 - 1 - 5 = 24.
The only way to get 24 with the remaining numbers is to use 7 and 17, or 9 and 15.
There are nine numbers and nine circles so each number must be used once.
Looking back at the possible combinations, the only one to use 11 is side 1,3, so one side must be 1,11,15,3.
Likewise the only side to use 13 is 3,5, so a second side must be
3,13,9,5.
This leaves only two numbers unused, making the third side:
1,7,17,5
Of course the order of the middle two numbers in each side doesn't matter, nor does the order in which the three sides are displayed.
Answer:
Step-by-step explanation:
The sum of all nine given numbers is 81.
The sum of all three sides of the triangle is 3 x 30 = 90
Three numbers in the corners will be counted twice in the sum of 90,
so the sum of the three numbers in the corners must be 90 - 81 = 9
The only way to get a sum of 9 with three of the given numbers is to take 1, 3 and 5. So those are the three numbers in the corners. It doesn't matter which of these three numbers gets put in each corner.
Consider the side with 1 and 3 at its ends: the sum of the remaining two circles must be 30 - 1 - 3 = 26.
The only way to get 26 with the remaining numbers is to use 17 and 9, or 11 and 15.
Consider the side with 3 and 5 at its ends: the sum of the remaining two circles must be 30 - 3 - 5 = 22.
The only way to get 22 with the remaining numbers is to use 7 and 15, or 9 and 13.
Consider the side with 1 and 5 at its ends: the sum of the remaining two circles must be 30 - 1 - 5 = 24.
The only way to get 24 with the remaining numbers is to use 7 and 17, or 9 and 15.
There are nine numbers and nine circles so each number must be used once.
Looking back at the possible combinations, the only one to use 11 is side 1,3, so one side must be 1,11,15,3.
Likewise the only side to use 13 is 3,5, so a second side must be
3,13,9,5.
This leaves only two numbers unused, making the third side:
1,7,17,5
Of course the order of the middle two numbers in each side doesn't matter, nor does the order in which the three sides are displayed.