Remark
This problem drove 4 of us crazy until one of us made a break through.
Assumptions.
He must eat either 1 pizza a day or 1 sandwich a day. He can't eat 2 of anything during 1 day. So he can't eat 2 pizzas in a day or 2 sandwiches in a day.
Step One
Find the number of days in a cycle.
3 out of 4 days means out of 4 days he chooses Pizza
2 out of 3 days means out of 3 days he chooses a Sandwich.
That means that there are 4 + 3 days in total = 7 days in the cycle.
Step Two
Find the number of Pizzas and the number of Sandwiches he had during the 42 day period.
3/7 * 42 = Pizzas
3*6 = 18 pizzas in 42 days.
4/7* 42 = Sandwiches
4*6 = 24 sandwiches.
In 42 days he ate 24 + 18 = 42 lunches of either pizzas or sandwiches.
Step three This is the key step.
Find the ratio of sandwiches and pizzas for x days.
Every 5 days he either has 3 pizzas or 2 sandwiches. If the total number of days is x then
3/5 * x = Pizzas
2/5 * x = Sandwiches.
Step 4
Equate the results.
Pizzas = sandwiches.
18 + 3/5 x = 24 + 2/5 x Multiply by 5
18*5 + (3/5)*x * 5 = 24 * 5 + (2/5)*x*5
90 + 3x = 120 + 2x Subtract 2x from both sides
90 + 3x - 2x = 120
90 + x = 120 Subtract 90 from both sides.
x = 120 - 90
x = 30 days.
Conclusion
It takes 30 days to even up the number of pizzas with the number of sandwiches.
batches. We will need 
cups of butter. This is 
cups. Since we can only buy butter in boxes with 2 cups of butter in it, we will have to buy 3. This will give us 6 cups of butter, which will be enough. If we only buy 2, then we will only have 4 cups of butter, which won't be enough to bake 15 batches of cookies.