O'Halloran numbers
For this task, for our purposes, a regular integer cuboid is a 3 dimensional rectangular object, with six faces, where all angles are right angles, where opposite faces of the cuboid are equal, and where each dimension is a positive integer unit length. It will subsequently be referred to simply as a cuboid; but be aware that it references the above definition.
The surface area of a cuboid is two times the length times the width, plus two times the length times the height, plus two times the width times the height. A cuboid will always have an even integer surface area. The minimum surface area a cuboid may have is 6; one where the l, w, and h measurements are all 1.
2 × ( l × w + w × h + h × l ) 2 × ( 1 × 1 + 1 × 1 + 1 × 1 ) = 6
Different cuboid configurations (may) yield different surface areas, but the surface area is always an integer and is always even.
A cuboid with l = 2, w = 1 h = 1 has a surface area of 10
2 × ( 2 × 1 + 1 × 1 + 1 × 2 ) = 10
There is no configuration which will yield a surface area of 8.
There are 16 known even integer values below 1000 which can not be a surface area for any integer cuboid. It is conjectured, though not rigorously proved, that no others exist.
- Task
- Find and display the even integer values that can not be the surface area of a regular, integer, rectangular, cuboid, larger than 6 (the minimum) and less than 1000.
- See also
Raku
my @Area;
my $threshold = 2000; # a little overboard to make sure we get them all
for 1..$threshold -> $x {
for 1..$x -> $y {
last if $x * $y > $threshold;
for 1..$y -> $z {
push @Area[($x × $y + $y × $z + $z × $x) × 2], "$x,$y,$z";
last if $x * $y * $z > $threshold;
}
}
}
say "Even integer surface areas NOT achievable by any regular, integer dimensioned cuboid:\n" ~
@Area[^$threshold].kv.grep( { $^key > 6 and $key %% 2 and !$^value } )»[0];
- Output:
Even integer surface areas NOT achievable by any regular, integer dimensioned cuboid: 8 12 20 36 44 60 84 116 140 156 204 260 380 420 660 924