Question

1. draw two straight lines across the clock to divide it into three parts whose numbers within add up to 26.

clock image with numbers 1 - 12

Explanation:

Step1: Calculate total sum of clock numbers

The numbers on a clock are 1 to 12. The sum of an arithmetic series is given by $\frac{n(n + 1)}{2}$, where $n = 12$. So, $\frac{12\times(12 + 1)}{2}=\frac{12\times13}{2}=78$. We need to divide this into three parts each summing to 26, so $3\times26 = 78$, which matches the total, so it's possible.

Step2: Find groups of numbers summing to 26

• First group: Let's try 11, 12, 1, 2. Sum: $11 + 12 + 1 + 2 = 26$.
• Second group: 9, 10, 3, 4. Sum: $9 + 10 + 3 + 4 = 26$.
• Third group: 5, 6, 7, 8. Sum: $5 + 6 + 7 + 8 = 26$.

Step3: Draw lines to separate these groups

Draw one line between 2 and 3, and another line between 4 and 5 (or equivalently, between 8 and 9, and between 10 and 11, depending on how you visualize the groups). Wait, actually, to separate the groups: the first group is 11,12,1,2; second is 9,10,3,4; third is 5,6,7,8. So draw a line from between 2 and 3 to between 10 and 11, and another line from between 4 and 5 to between 8 and 9. This will divide the clock into three parts: one with 11,12,1,2; one with 9,10,3,4; one with 5,6,7,8.

Answer:

Draw one line between the 2 - 3 area and 10 - 11 area, and another line between the 4 - 5 area and 8 - 9 area (or equivalent to separate the groups 11,12,1,2; 9,10,3,4; 5,6,7,8 each summing to 26). The visual division would have the first part with 11,12,1,2; second with 9,10,3,4; third with 5,6,7,8.