The problem involves a triangle $ABC$ where:

1. Angle $A$ is three times the measure of angle $B$.
2. Angle $C$ is twice the measure of angle $B$.

Let $B = x$ (in degrees).

Then:

• $A = 3x$
• $C = 2x$

Since the sum of the angles in any triangle is $180^\circ$, we can write the equation:

$A + B + C = 180^\circ$

Substituting the values:

$3x + x + 2x = 180^\circ$

This simplifies to:

$6x = 180^\circ$

Dividing both sides by 6:

$x = 30^\circ$

Now, we can find the measures of all angles:

• $B = x = 30^\circ$
• $A = 3x = 3 \times 30 = 90^\circ$
• $C = 2x = 2 \times 30 = 60^\circ$

The smallest angle in the triangle is $B = 30^\circ$.

Answer: The measure found by the artist for the smallest angle is $30^\circ$.

Would you like more details on the steps or have any questions?

Here are some related questions to consider:

1. How does the sum of angles change in a non-Euclidean triangle?
2. What would happen if angle $C$ was four times $B$?
3. Can this problem be solved using trigonometric identities?
4. What other relationships between angles can form a right triangle?
5. How can this approach be applied to polygons with more than three sides?

Tip: When working with angles in a triangle, remember that their sum is always $180^\circ$. This property is fundamental in geometry and helps in solving many angle-based problems.