Question

Two triangles are considered the same if they have the same three side lengths. Find all possible values of P such that there is only 1 possible triangle with integer side lengths and perimeter P.

Answer 1

P ∈ {3, 5, 6, 8}

Explanation:

You want the possible perimeter values such that there can only be one triangle with that perimeter and integer side lengths.

Conditions

The requirement that the triangle satisfies the triangle inequality means the longest side must be shorter than P/2.

Trials

The smallest triangle with integer side lengths is the {1, 1, 1} equilateral triangle.

P=3, {1, 1, 1}

For P=4, the longest side can be neither 1 nor 2, so P=4 is not an option.

For P=5, the longest side can be 2, requiring the other sides to be 2 and 1:

P = 5, {2, 2, 1}

For P=6, the longest side must be less than 3, so all sides must be 2.

P = 6, {2, 2, 2}

For P=7, two triangles are possible: {3, 3, 1} and {3, 2, 2}.

For P = 8, the longest side must be less than 4, so will be 3. The other two sides must be 3 and 2.

P = 8, {3, 3, 2}

For all values of P above 8, there are at least two possible triangles.

Possible values of P are 3, 5, 6, 8.