Scalenia is a country bounded by
three straight frontiers called A, B,
and C, of course, each of a different
length but each an exact number of
kilometers long.
The curious thing is
that the length of A+B+C, and
the lengths of A+B-C, A+C-B, and
B+C-A, are all precisely square
numbers of kilometers.
If Scalenia
has the smallest perimeter consistent with this curious fact, what is
the length of each of its frontiers?
clearvars
ct=0; clc
for sra=3:2000
tot=sra^2;
for a=1:tot/3
for b=a+1:(tot-a)/2
c=tot-a-b;
if c<b+a
sr=round(sqrt(a+b-c));
if sr*sr==a+b-c
sr=round(sqrt(c+a-b));
if sr*sr==c+a-b
sr=round(sqrt(c+b-a));
if sr*sr==c+b-a
fprintf('%5d %5d %5d %10d%7d\n',a, b, c, tot, sra)
fprintf('%5d',a+b-c,a+c-b,b+c-a)
fprintf('\n',tot)
fprintf('%5d',sqrt(a+b-c),sqrt(a+c-b),sqrt(b+c-a))
fprintf('\n\n')
ct=ct+1;
if ct==10
return
end
end
end
end
end
end
end
end
finds the first ten such cases (10 lowest perimeters):
The first line in each set shows A, B and C, in size order, then the perimeter and its square root.
The second line shows the three differences asked for.
The third line shows the respective square roots of those differences.
26 80 90 196 14
16 36 144
4 6 12
34 130 160 324 18
4 64 256
2 8 16
64 130 130 324 18
64 64 196
8 8 14
80 170 234 484 22
16 144 324
4 12 18
144 170 170 484 22
144 144 196
12 12 14
50 306 320 676 26
36 64 576
6 8 24
104 320 360 784 28
64 144 576
8 12 24
58 400 442 900 30
16 100 784
4 10 28
208 250 442 900 30
16 400 484
4 20 22
290 416 450 1156 34
256 324 576
16 18 24
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Posted by Charlie
on 2025-02-04 09:12:11 |
computer-aided solution (spoiler)
