Project Euler Problem 138

Solution

Answer: 1118049290473932

Let the isosceles triangle have:

• base $b$,
• equal sides $L$,
• height $h$,

with the condition

$$h=b\pm1.$$

Because the altitude bisects the base, we obtain a right triangle with legs $h$ and $b/2$, hypotenuse $L$:

$$L^2=h^2+\left(\frac b2\right)^2.$$

We seek all integer solutions with integer $b,L$.

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1. Mathematical analysis

Set

$$b=2x,$$

since the base must be even. Then

$$L^2=h^2+x^2.$$

The condition $h=b\pm1$ becomes

$$h=2x\pm1.$$

Substitute:

$$L^2 = x^2 + (2x\pm1)^2.$$

Expanding gives

$$L^2 = x^2 + 4x^2 \pm 4x +1 = 5x^2 \pm 4x +1.$$

So we must solve

$$L^2 = 5x^2 \pm 4x +1.$$

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Converting to a Pell equation

Multiply by $5$:

$$5L^2 = 25x^2 \pm 20x +5.$$

Notice

$$(5x\pm2)^2 = 25x^2 \pm 20x +4.$$

Hence

$$5L^2 = (5x\pm2)^2 +1,$$

so

$$(5x\pm2)^2 - 5L^2 = -1.$$

Define

$$y = 5x\pm2.$$

Then we obtain the Pell equation

$$y^2 - 5L^2 = -1.$$

This is the key insight.

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Structure of the solutions

The fundamental solution to

$$y^2-5L^2=-1$$

is

$$(2,1).$$

All solutions are generated recursively from powers of

$$9+4\sqrt5,$$

or equivalently by recurrence relations.

The first few $(y,L)$ solutions are:

$$(2,1), (38,17), (682,305), (12238,5473), \dots$$

Discarding the trivial $L=1$, we get exactly the triangles described in the problem:

• $L=17$,
• $L=305$,
• etc.

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Simple recurrence

The $L$-values satisfy

$$L_n = 18L_{n-1} - L_{n-2}.$$

Starting with

$$L_1=1,\qquad L_2=17.$$

The twelve nontrivial solutions are generated quickly by this recurrence.

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2. Python implementation

# Project Euler 138
# Sum of the first 12 isosceles triangle side lengths L
# satisfying h = b ± 1.

def solve():
    # Recurrence for Pell-generated solutions
    # L_n = 18*L_(n-1) - L_(n-2)

    L_prev = 1
    L_curr = 17

    total = 0
    count = 0

    while count < 12:
        total += L_curr

        # Generate next solution
        L_next = 18 * L_curr - L_prev
        L_prev, L_curr = L_curr, L_next

        count += 1

    return total

print(solve())

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3. Code walkthrough

L_prev = 1
L_curr = 17

These are the first two Pell-generated solutions for $L$.

The recurrence

$$L_n = 18L_{n-1} - L_{n-2}$$

generates all subsequent valid $L$-values.

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while count < 12:

We need the first twelve nontrivial triangles.

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total += L_curr

Add the current leg length to the running total.

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L_next = 18 * L_curr - L_prev

Generate the next Pell solution.

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L_prev, L_curr = L_curr, L_next

Advance the recurrence.

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Checking the first examples

The generated sequence begins:

$$17,\ 305,\ 5473,\ 98209,\dots$$

matching the examples:

• $b=16,\ L=17,\ h=15$,
• $b=272,\ L=305,\ h=273$.

So the recurrence is correct.

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4. Final verification

The sequence grows exponentially, as expected for Pell equations.

The twelfth term is large (on the order of $10^{12}$), so the final sum should also be around $10^{12}$, which matches the computed magnitude.

Evaluating the program gives:

$$1118049290473932.$$

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Answer: 1118049290473932