A number theory problem by A Former Brilliant Member
Let be the number of noncongruent triangles having the perimeter . Find the ratio of
The answer is 1.00.
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1 solution
f(2n+1) is always equal to f(2n+4), therefore the ratio is always 1.
To prove this, I computed all noncongruent triangles with integral side lengths and a max side length of 50, there are 8924 of these.
Then kept an array and count of all of the different perimeters we found. This would represent f(n) for this question.
After looking through that array, I noticed things like:
When n=5, f(11)=1, f(14)=1, 1/1=1
When n=10, f(21)=7, f(24)=7, 7/7=1
When n=15, f(31)=16, f(34)=16, 16/16=1
When n=20, f(41)=30, f(44)=30, 30/30=1
When n=25, f(51)=48, f(54)=48, 48/48=1
When n=30, f(61)=70, f(64)=70, 70/70=1
When n=35, f(71)=96, f(74)=96, 96/96=1
When n=40, f(81)=127, f(84)=127, 127/127=1
This shows pretty solidly that the answer should be 1, good observation and good problem!