Solution:

Let's break down the given information step by step to find the solution to the problem.

Given information:
- 20 men can do the task in 12 days working for 8 hours a day.
- 24 women can do the task in 12 days working for 8 hours a day.
- 40 boys can do the task in 12 days working for 8 hours a day.

Finding the work rate:
To find the work rate of each group, we need to calculate the work done by each group in a day.

Work done by 20 men in a day = 1/12 (total work)
Work done by 1 man in a day = (1/12) / 20 = 1/240 (work rate of 1 man)

Similarly,
Work done by 24 women in a day = 1/12 (total work)
Work done by 1 woman in a day = (1/12) / 24 = 1/288 (work rate of 1 woman)

And,
Work done by 40 boys in a day = 1/12 (total work)
Work done by 1 boy in a day = (1/12) / 40 = 1/480 (work rate of 1 boy)

Calculating the total work done by the given group:
- 6 women and 2 boys will be working together with some men.
- Let's assume x men are working in this group.

Total work done by this group in a day = (6 * 1/288 + 2 * 1/480 + x * 1/240)

Calculating the total work done by this group in 12 days:
Total work done by this group in 12 days = 12 * (6 * 1/288 + 2 * 1/480 + x * 1/240)

Calculating the total work done by this group in 5 hours a day for 12 days:
Total work done by this group in 12 days = (5/8) * 12 * (6 * 1/288 + 2 * 1/480 + x * 1/240)

Finding the number of men required:
Since the total work is four times bigger than the previous task, we can set up the following equation:

(5/8) * 12 * (6 * 1/288 + 2 * 1/480 + x * 1/240) = 4 * 1/12

Simplifying the equation:
(5/8) * 12 * (6/288 + 2/480 + x/240) = 4/12

(5/8) * 12 * (20/960 + 16/960 + x/240) = 1/3

(5/8) * 12 * (36/960 + x/240) = 1/3

(5/8) * 12 * (36 + 4x) = 1/3

(5/8) * 12 * 36 + (5/8) * 12 * 4x = 1/3

(5/8) * 432 + (5/8) * 48x = 1/3

216 + 30x = 16/3

30x = 16/3 -