1. The problem involves understanding the relationship between the numbers in the diamond shapes and the rectangles above them. 2. Each diamond is divided into two triangles with numbers, and there is a rectangle above with another number. 3. We need to find the rule connecting the two triangle numbers to the rectangle number. 4. For the first diamond: triangles are 2 and 6, rectangle is 4. 5. For the second diamond: triangles are 7 and 9, rectangle is 21. 6. For the third diamond: triangles are 5 and 6, rectangle is 10. 7. Let's test if the rectangle number is the product of the two triangle numbers: - $2 \times 6 = 12$, but rectangle is 4, so no. - $7 \times 9 = 63$, but rectangle is 21, so no. - $5 \times 6 = 30$, but rectangle is 10, so no. 8. Let's test if the rectangle number is the sum of the two triangle numbers: - $2 + 6 = 8$, rectangle is 4, no. - $7 + 9 = 16$, rectangle is 21, no. - $5 + 6 = 11$, rectangle is 10, no. 9. Let's test if the rectangle number is the difference of the two triangle numbers: - $6 - 2 = 4$, rectangle is 4, yes. - $9 - 7 = 2$, rectangle is 21, no. - $6 - 5 = 1$, rectangle is 10, no. 10. Let's test if the rectangle number is the product divided by the sum: - $\frac{2 \times 6}{2 + 6} = \frac{12}{8} = 1.5$, rectangle is 4, no. - $\frac{7 \times 9}{7 + 9} = \frac{63}{16} = 3.9375$, rectangle is 21, no. - $\frac{5 \times 6}{5 + 6} = \frac{30}{11} = 2.727$, rectangle is 10, no. 11. Let's test if the rectangle number is the product divided by one of the triangle numbers: - $\frac{2 \times 6}{3} = 4$ if 3 is the average of 2 and 6, but 3 is not average. 12. Let's test if the rectangle number is the product divided by the other triangle number: - $\frac{2 \times 6}{6} = 2$, rectangle is 4, no. 13. Let's test if the rectangle number is the product divided by the smaller triangle number: - $\frac{2 \times 6}{2} = 6$, rectangle is 4, no. 14. Let's test if the rectangle number is the product divided by the larger triangle number: - $\frac{2 \times 6}{6} = 2$, rectangle is 4, no. 15. Let's test if the rectangle number is the product divided by 3: - $\frac{2 \times 6}{3} = 4$, rectangle is 4, yes. - $\frac{7 \times 9}{3} = 21$, rectangle is 21, yes. - $\frac{5 \times 6}{3} = 10$, rectangle is 10, yes. 16. So the rule is: rectangle number = $\frac{\text{left triangle} \times \text{right triangle}}{3}$. Final answer: The rectangle number equals the product of the two triangle numbers divided by 3, i.e., $$\text{Rectangle} = \frac{\text{Left} \times \text{Right}}{3}$$.