My friend , WG challenged me as follows:
״ I wrote down 5 positive, not necessarily distinct integers and evaluated all possible pairwise sums.
The result was -
three distinct values: s1;s2;& s3.
Can you evaluate my initial five?”
I did and now I pose a question to you:
What relationship between s1 s2 & s3 enables a viable and unique solution to WG’s original puzzle?
If the three values from low to high are odd, even, odd, then it's also they will be expressible as 2n-k, 2n, 2n+k.
In this case, the original five numbers are n-k, three n's, and n+k. For example if the three sums were 5, 8 and 11, the original five numbers were 1, 4, 4, 4 and 7.
The same can't be done when the highest and lowest numbers are even, such as 18, 21, 24, which could result from 9, 9, 9, 12, 12 or 9, 9, 12, 12, 12.
This observation comes from trying all sets of 5 using all such combinations from the numbers 1 through 12. In the below output the first of two rows gives the set of five numbers and the number of values present in those five (most are a "full house" of three and two, but some are three different values: three of one and one each of another two). But all have that 3 of a kind.
The ones that meet the criteria for being able to determine the original five are marked with *** below the set.
The relationship: when in order, s1, s2, s3, the first (lowest) and last(highest) are odd, and the middle one is the average of the two outer ones.
--- values--- number of
different
values
1 1 1 2 2 2
2 3 4 the 3 sums
1 1 1 3 3 2
2 4 6
1 1 1 4 4 2
2 5 8
1 1 1 5 5 2
2 6 10
1 1 1 6 6 2
2 7 12
1 1 1 7 7 2
2 8 14
1 1 1 8 8 2
2 9 16
1 1 1 9 9 2
2 10 18
1 1 1 10 10 2
2 11 20
1 1 1 11 11 2
2 12 22
1 1 1 12 12 2
2 13 24
1 1 2 2 2 2
2 3 4
1 1 3 3 3 2
2 4 6
1 1 4 4 4 2
2 5 8
1 1 5 5 5 2
2 6 10
1 1 6 6 6 2
2 7 12
1 1 7 7 7 2
2 8 14
1 1 8 8 8 2
2 9 16
1 1 9 9 9 2
2 10 18
1 1 10 10 10 2
2 11 20
1 1 11 11 11 2
2 12 22
1 1 12 12 12 2
2 13 24
1 2 2 2 3 3
3 4 5 largest of 3 sums, 2+3 = 5, which is odd
***
1 3 3 3 5 3
4 6 8
1 4 4 4 7 3 ditto 4+7, etc.
5 8 11
***
1 5 5 5 9 3
6 10 14
1 6 6 6 11 3
7 12 17 6+11
***
2 2 2 3 3 2
4 5 6
2 2 2 4 4 2
4 6 8
2 2 2 5 5 2
4 7 10
2 2 2 6 6 2
4 8 12
2 2 2 7 7 2
4 9 14
2 2 2 8 8 2
4 10 16
2 2 2 9 9 2
4 11 18
2 2 2 10 10 2
4 12 20
2 2 2 11 11 2
4 13 22
2 2 2 12 12 2
4 14 24
2 2 3 3 3 2
4 5 6
2 2 4 4 4 2
4 6 8
2 2 5 5 5 2
4 7 10
2 2 6 6 6 2
4 8 12
2 2 7 7 7 2
4 9 14
2 2 8 8 8 2
4 10 16
2 2 9 9 9 2
4 11 18
2 2 10 10 10 2
4 12 20
2 2 11 11 11 2
4 13 22
2 2 12 12 12 2
4 14 24
2 3 3 3 4 3
5 6 7
***
2 4 4 4 6 3
6 8 10
2 5 5 5 8 3
7 10 13
***
2 6 6 6 10 3
8 12 16
2 7 7 7 12 3
9 14 19
***
3 3 3 4 4 2
6 7 8
3 3 3 5 5 2
6 8 10
3 3 3 6 6 2
6 9 12
3 3 3 7 7 2
6 10 14
3 3 3 8 8 2
6 11 16
3 3 3 9 9 2
6 12 18
3 3 3 10 10 2
6 13 20
3 3 3 11 11 2
6 14 22
3 3 3 12 12 2
6 15 24
3 3 4 4 4 2
6 7 8
3 3 5 5 5 2
6 8 10
3 3 6 6 6 2
6 9 12
3 3 7 7 7 2
6 10 14
3 3 8 8 8 2
6 11 16
3 3 9 9 9 2
6 12 18
3 3 10 10 10 2
6 13 20
3 3 11 11 11 2
6 14 22
3 3 12 12 12 2
6 15 24
3 4 4 4 5 3
7 8 9
***
3 5 5 5 7 3
8 10 12
3 6 6 6 9 3
9 12 15
***
3 7 7 7 11 3
10 14 18
4 4 4 5 5 2
8 9 10
4 4 4 6 6 2
8 10 12
4 4 4 7 7 2
8 11 14
4 4 4 8 8 2
8 12 16
4 4 4 9 9 2
8 13 18
4 4 4 10 10 2
8 14 20
4 4 4 11 11 2
8 15 22
4 4 4 12 12 2
8 16 24
4 4 5 5 5 2
8 9 10
4 4 6 6 6 2
8 10 12
4 4 7 7 7 2
8 11 14
4 4 8 8 8 2
8 12 16
4 4 9 9 9 2
8 13 18
4 4 10 10 10 2
8 14 20
4 4 11 11 11 2
8 15 22
4 4 12 12 12 2
8 16 24
4 5 5 5 6 3
9 10 11
***
4 6 6 6 8 3
10 12 14
4 7 7 7 10 3
11 14 17
***
4 8 8 8 12 3
12 16 20
5 5 5 6 6 2
10 11 12
5 5 5 7 7 2
10 12 14
5 5 5 8 8 2
10 13 16
5 5 5 9 9 2
10 14 18
5 5 5 10 10 2
10 15 20
5 5 5 11 11 2
10 16 22
5 5 5 12 12 2
10 17 24
5 5 6 6 6 2
10 11 12
5 5 7 7 7 2
10 12 14
5 5 8 8 8 2
10 13 16
5 5 9 9 9 2
10 14 18
5 5 10 10 10 2
10 15 20
5 5 11 11 11 2
10 16 22
5 5 12 12 12 2
10 17 24
5 6 6 6 7 3
11 12 13
***
5 7 7 7 9 3
12 14 16
5 8 8 8 11 3
13 16 19
***
6 6 6 7 7 2
12 13 14
6 6 6 8 8 2
12 14 16
6 6 6 9 9 2
12 15 18
6 6 6 10 10 2
12 16 20
6 6 6 11 11 2
12 17 22
6 6 6 12 12 2
12 18 24
6 6 7 7 7 2
12 13 14
6 6 8 8 8 2
12 14 16
6 6 9 9 9 2
12 15 18
6 6 10 10 10 2
12 16 20
6 6 11 11 11 2
12 17 22
6 6 12 12 12 2
12 18 24
6 7 7 7 8 3
13 14 15
***
6 8 8 8 10 3
14 16 18
6 9 9 9 12 3
15 18 21
***
7 7 7 8 8 2
14 15 16
7 7 7 9 9 2
14 16 18
7 7 7 10 10 2
14 17 20
7 7 7 11 11 2
14 18 22
7 7 7 12 12 2
14 19 24
7 7 8 8 8 2
14 15 16
7 7 9 9 9 2
14 16 18
7 7 10 10 10 2
14 17 20
7 7 11 11 11 2
14 18 22
7 7 12 12 12 2
14 19 24
7 8 8 8 9 3
15 16 17
***
7 9 9 9 11 3
16 18 20
8 8 8 9 9 2
16 17 18
8 8 8 10 10 2
16 18 20
8 8 8 11 11 2
16 19 22
8 8 8 12 12 2
16 20 24
8 8 9 9 9 2
16 17 18
8 8 10 10 10 2
16 18 20
8 8 11 11 11 2
16 19 22
8 8 12 12 12 2
16 20 24
8 9 9 9 10 3
17 18 19
***
8 10 10 10 12 3
18 20 22
9 9 9 10 10 2
18 19 20
9 9 9 11 11 2
18 20 22
9 9 9 12 12 2
18 21 24
9 9 10 10 10 2
18 19 20
9 9 11 11 11 2
18 20 22
9 9 12 12 12 2
18 21 24
9 10 10 10 11 3
19 20 21
***
10 10 10 11 11 2
20 21 22
10 10 10 12 12 2
20 22 24
10 10 11 11 11 2
20 21 22
10 10 12 12 12 2
20 22 24
10 11 11 11 12 3
21 22 23
***
11 11 11 12 12 2
22 23 24
11 11 12 12 12 2
22 23 24
clc,clearvars
ix=combinator(5,2,'c');
for a=1:12
for b=a:12
for c=b:12
for d=c:12
for e=d:12
set=[a b c d e];
setu=unique(set);
tots=[];
for i=1:length(ix)
tots(end+1)=sum(set(ix(i,:)));
end
tots=unique(tots);
if length(tots)==3
fprintf('%2d %2d %2d %2d %2d %2d\n',set,length(setu))
disp(tots)
if mod(max(tots),2)==1
disp('***')
else
disp(' ')
end
end
end
end
end
end
end
|
|
Posted by Charlie
on 2023-02-28 15:37:26 |
No Subject
