Question 62

Let $N = 6$ be the total number of people seated around the circular table. Let $W$ be the number of women and $M$ be the number of men. We have:

Given constraints:

1. At least two men and two women: $W \ge 2$ and $M \ge 2$, so $W \in \{2, 3, 4\}$.
2. "None of the women are right-handed": Every person is either right-handed ($R$) or left-handed ($L$). Since no woman is right-handed, all women must be left-handed ($W \subseteq L$).
3. "At least three right-handed persons": Since all right-handed persons must be men, the number of right-handed men must be at least 3 ($M_R \ge 3$). Consequently, $M \ge 3$.
   From $W + M = 6$ and $M \ge 3$, we deduce $W \le 3$. Combining with $W \ge 2$, we have $W \in \{2, 3\}$.

Test $W = 3$:

• If $W = 3$, then $M = 3$.
• Since $M_R \ge 3$ and $M = 3$, all 3 men must be right-handed ($M_R = 3$, $M_L = 0$).
• Thus, the left-handed people at the table are precisely the 3 women, and no one else.
• Now apply the condition: "Every woman has a left-handed person to her immediate right."
  Since only women are left-handed, to the immediate right of every woman there must be another woman.
  Starting from any woman and moving clockwise to the right, we must immediately encounter a woman. Following this around the circle would require every person in the circle to be a woman, which is impossible because there are 3 men at the table.
  Specifically, any contiguous block of women must end with a woman who has a man to her right, violating the condition.
  Hence, $W = 3$ is impossible.

Test $W = 2$:

• If $W = 2$, then $M = 4$.
• To have $M_R \ge 3$, there are either 3 or 4 right-handed men. If there are 3 right-handed men, exactly 1 man is left-handed ($M_L = 1$).
• The left-handed people are the 2 women and 1 man, totaling 3 left-handed people.
• We can easily arrange them so that every woman has a left-handed person to her immediate right:
  For instance, moving clockwise around the circle:
  $$\text{Woman}_1 (L) \to \text{Woman}_2 (L) \to \text{Man}_1 (L) \to \text{Man}_2 (R) \to \text{Man}_3 (R) \to \text{Man}_4 (R)$$
  Here:
  • To the immediate right of $\text{Woman}_1$ is $\text{Woman}_2$ (left-handed).
  • To the immediate right of $\text{Woman}_2$ is $\text{Man}_1$ (left-handed).
  • Both women have a left-handed person to their immediate right.
  • There are 2 women, 4 men, and 3 right-handed people.
    All conditions are fully satisfied.

Therefore, the number of women at the table must be exactly 2.