Find a rectangular prism whose edges and main diagonal have integer lengths, such that the squares of the edges are 10-digit pandigital numbers.
I'm assuming this is a box and the main diagonal is from one corner to the opposite corner.
I'm also assuming that the edge lengths are not necessarily distinct.
Let the edge lengths be integers a,b,c s.t. each of these when squared forms a pandigital 10-digit integer. Also √(a^2 + b^2 + c^2) must be an integer
There are only 87 10-digit pandigitals which are perfect squares.
Their square roots are the possible values for edge lengths.
Searching a combination of those 87 values taken 3 at a time (I used combinations with replacement since there is no requirement that the 3 different edge lengths be distinct), I looked for integer diagonal values.
I found only one solution, which in fact, did have 2 of the lengths equal
side 1 side 2 side 3 diagonal
45624 91248 91248 136872.0
The squares of each side is indeed a 10-digit pandigital:
2081549376 8326197504 8326197504
Check, since diagonal calc used an "epsilon" method:
45624^2 + 91248^2 + 91248^2 = 18733944384
136872^2 = 18733944384
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def intdiag(a,b,c):
epsilon = .0000001
diag = (a**2 + b**2 + c**2)**.5
if abs(diag - round(diag)) < epsilon:
return True
return False
def issquare(n):
""" Input an integer, Returns True iff it is a perfect square. """
if round(n**0.5)**2 == n:
return True
else:
return False
edgeworthies = []
from itertools import permutations
from itertools import combinations_with_replacement
for p in permutations('0123456789'):
if p[0] == '0':
continue
n = int(''.join(p))
if issquare(n):
edgeworthies.append(int(round(n**.5)))
for comb in combinations_with_replacement(edgeworthies,3):
a = comb[0]
b = comb[1]
c = comb[2]
if intdiag(a,b,c):
diag = (a**2 + b**2 + c**2)**.5
print(a,b,c,diag)
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Posted by Larry
on 2023-10-18 14:18:16 |
Computer solution
