1. **Problem Statement:** You need to fill four boxes with digits 1 to 9 (each digit used at most once) in three different ways: - First, so exactly two triangles exist. - Second, so exactly one triangle exists. - Third, so no triangle exists. 2. **Understanding the Problem:** The problem involves checking if three side lengths can form a triangle. The triangle inequality states: $$a + b > c, \quad b + c > a, \quad c + a > b$$ where $a$, $b$, and $c$ are the side lengths. 3. **Assigning Variables:** Let the four boxes be $A$, $B$, $C$, and $D$. Assuming the large triangle sides are formed by pairs of these numbers: - Triangle 1 sides: $A$, $B$, $C$ - Triangle 2 sides: $A$, $B$, $D$ - Triangle 3 sides: $A$, $C$, $D$ 4. **Goal:** - For exactly two triangles: two of these triples satisfy the triangle inequality. - For exactly one triangle: only one triple satisfies it. - For no triangle: none satisfy it. 5. **Example Solutions:** **Exactly two triangles:** Choose $A=3$, $B=4$, $C=5$, $D=8$ - Triangle 1: $3,4,5$ satisfies $3+4>5$, $4+5>3$, $5+3>4$ (valid) - Triangle 2: $3,4,8$ check $3+4>8$? No (7>8 false), invalid - Triangle 3: $3,5,8$ check $3+5>8$? No (8>8 false), invalid Only one triangle here, so adjust $D$. Try $D=6$: - Triangle 2: $3,4,6$ check $3+4>6$? Yes (7>6) - Triangle 3: $3,5,6$ check $3+5>6$? Yes (8>6) So triangles 2 and 3 valid, triangle 1 valid too, so all three triangles exist. Try $D=7$: - Triangle 2: $3,4,7$ check $3+4>7$? No (7>7 false) - Triangle 3: $3,5,7$ check $3+5>7$? Yes (8>7) So triangles 1 and 3 valid, triangle 2 invalid, exactly two triangles. **Exactly one triangle:** Choose $A=2$, $B=3$, $C=8$, $D=9$ - Triangle 1: $2,3,8$ check $2+3>8$? No (5>8 false) - Triangle 2: $2,3,9$ check $2+3>9$? No (5>9 false) - Triangle 3: $2,8,9$ check $2+8>9$? Yes (10>9 true) Only triangle 3 exists. **No triangle:** Choose $A=1$, $B=2$, $C=8$, $D=9$ - Triangle 1: $1,2,8$ check $1+2>8$? No (3>8 false) - Triangle 2: $1,2,9$ check $1+2>9$? No (3>9 false) - Triangle 3: $1,8,9$ check $1+8>9$? No (9>9 false) No triangles exist. 6. **Final answers:** - Exactly two triangles: $A=3$, $B=4$, $C=5$, $D=7$ - Exactly one triangle: $A=2$, $B=3$, $C=8$, $D=9$ - No triangle: $A=1$, $B=2$, $C=8$, $D=9$