Leilani placed a digit in each of these 16 squares:
+------+------+------+------+
| | | | |
+---- -+------+------+------+
| | | | |
+------+------+------+------+
| | | | |
+------+------+------+------+
| | | | |
+------+------+------+------+
When she had finished, the grid had the following properties:
• No digit occured more than once in any row.
• The sum of the four digits in each row was the same.
• The sum of the four digits in each column was the same.
• Each row formed a different four-digit perfect square.
Complete the 4x4 square grid given above.
Note: Adapted from Enigma #1553 which appeared in 'New Scientist' in 2009.
clearvars,clc
for i=32:99
sq(i,:)=char(string(i^2));
sm(i,:)=sum(sq(i,:)-'0');
end
[sums, idx]=sort(sm);
squares=sq(idx,:);
for i=1:length(sums)
fprintf('%s %5d\n',squares(i,:),sums(i))
end
for htot=7:25
idx=find(sums==htot);
if length(idx)>=4
sourceSq=squares(idx,:);
combins=combinator(length(sourceSq),4,'c');
for i=1:size(combins,1)
grid=sourceSq(combins(i,:),:);
good=true;
for col=1:4
s=sum(grid(:,col)-'0');
if col>1 && s~=prev
good=false;
break;
end
prev=s;
end
if good
disp(grid)
disp(' ')
end
end
end
end
find first the 4-digit squares ordered by their sod's:
1024 7
1600 7
2401 7
2500 7
1521 9
2025 9
2304 9
2601 9
3600 9
8100 9
1225 10
2116 10
3025 10
5041 10
6400 10
1156 13
1444 13
2209 13
2704 13
3136 13
3721 13
4225 13
4900 13
6241 13
1681 16
3364 16
3481 16
4624 16
7225 16
9025 16
1089 18
1296 18
1764 18
2916 18
3249 18
4356 18
4761 18
5184 18
5625 18
6084 18
6561 18
7056 18
9216 18
9801 18
1369 19
1936 19
2809 19
3844 19
4096 19
5329 19
6724 19
7921 19
8281 19
9604 19
1849 22
5476 22
7744 22
8464 22
9409 22
4489 25
5776 25
5929 25
7396 25
8836 25
3969 27
7569 27
8649 27
6889 31
and then searches the successive sod's for sets of four which satisfy the equal-column-total requirement, and finds
1764
3249
5184
9801
As is necessary, the equal column sums match the equal row sums.
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Posted by Charlie
on 2023-07-21 10:32:19 |
computer solution
