) Fun time. ( Brainstorming/ HOTS).
Instructions: a) You can use the numbers from 1 to 19 only. b) There are three dots marked on each line segment in the figure given. You have to replace the dots with the numbers such that sum of the
numbers on each side of the triangles must be 22 .
Answers
Given : numbers from 1 to 19 only.
To Find : replace the dots with the numbers such that sum of the
numbers on each side of the triangles must be 22.
Solution:
let say sum of central dots one external line = m
sum of all six side of triangle = 22 * 6 = 132
n is center of hexagon
132 + m - 5n = (19 * 20)/2
=> 132 + m - 5n = 190
=> m = 58 + 5n
Sum of all external line = 22 * 6 = 132
132 = m + 2 ( n)
p = sum of six corner dots of hexagonal
=> 2p = 132 - m ( as corner dots of hexagonal each dot is counted twice )
=> 2p = 132 - (58 + 5n)
=> 2p = 74 - 5n
hence n must be even
n = 2 , 4 or 6 can only satisfy this
as n = 8 => p = 17 and p is sum of 6 dots which can be minimum
1 + 2 + 3 + 4 + 5 + 6 = 21
So Lets try few solution with n = 2 , 4 or 6
n = 6 Does not give any solution as central dot on external line is fixed as 1 , 2 , 3 , 4 , 5 , 7 and leading to no final solution
n = 2 => p = 37 m = 68 as 2 is in center so 18 can not be in any central line applying such conditions and using hit and trial
for : n = 2 one of possible solution :
1 18 3
13 19 17 14
8 12 2 15 5
10 16 9 6
4 7 11
one More with n = 2
1 18 3
14 19 17 15
7 13 2 16 4
10 11 12 6
9 5 8
another one with n = 2
1 18 3
9 19 17 14
12 8 2 15 5
6 16 13 10
4 11 7
n = 4 => p = 27 m = 78 as 4 is in center so 14 can not be in any central line applying such conditions and using hit and trial
n = 4 one of possible solution :
10 9 3
7 8 15 18
5 13 4 17 1
11 12 16 19
6 14 2
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