You are looking for 6 numbers that correspond to the blacked out values in the recipe.
The Kitchen Clue #2
Use #3 and #10 together to figure out the number of eggs (E) and the ounces of butter (B). E = B and E2 = B. So, E=1 and B=1.
The Kitchen Clue #3
Use #2, #4 and the fact that you know eggs (E) is 1 and butter (B) is 1 to figure out the ounces of flour (F).
F is a perfect cube between 1 and 20, inclusive. There are two options for F: 1 (13 = 1) and 8 (23 = 8).
There are five distinct amounts and six missing values, so we know that only two of the values are equal. Since E=1 and B=1 already, F cannot be 1. So, F=8.
The Kitchen Clue #4
Use #5 to begin narrowing down the possibilities for the number of hours.
Flour (F) and hours (H) share two perfect squares as a common factor. Perfect squares are 1 (12 = 1), 4 (22 = 4), 9 (32 = 9), 16 (42 = 16), etc. Since F=8, the common factors that are perfect squares must be 1 and 4. So, we know that H has 4 as a common factor, which means H could be 4, 8, 12, 16, or 20. Since H is also the largest amount and F=8, H must be 12, 16, or 20.
The Kitchen Clue #5
Use #8 and #9 to narrow possibilities for sugar (S).
Because all amounts are between 1-20, you can get a quick minimum and maximum sum that the amounts can be. You should get only two possible numbers for the ounces of sugar that fit the sum's range.
We know F=8, B=1, and E=1.
First let's figure out what the minimum sum could be. Without using any other constraints except the fact that there are five distinct amounts, that would be if H=12, CS=2, and S (sugar) = 3. So the smallest possible sum is 8 + 1 + 1 + 12 + 2 + 3 = 27.
Next let's figure out the maximum sum. That would be if H=20, CS=19, and S=18. So the largest possible sum is 8 + 1 + 1 + 20 + 19 + 18 = 67.
Using #9, we know that S x F must be between 27 to 67, the possible sums. Since F=8, S could be 4, 5, 6, or 7 (S can't be 8 because F=8 already and we can't have any more duplicate amounts).
But we also know S is a prime number from #8, so S can only be 5 or 7.
The Kitchen Clue #6
Now using #7, you can guess and check to find the missing amounts.
The difference between H and CS is the same as the difference between F and S. F=8 and S could be 5 or 7. So the differences could be 3 (if S=5 then 8-5 = 3) or 1 (if S=7 then 8-7 = 1).
We will guess and check with both cases. Remember from Clue #4 that H could be 12, 16, or 20.
Trying S=5 so H - CS = 3 If H=12 then CS=9. We know that F=8, E=1, and B=1. We are assuming S=5. The sum of all the amounts is 36. S x F = 5 x 8 = 40. Since 36 does not equal 40, this combination is not correct.
If H=16 then CS=13. The sum of the amounts is 44. Since 44 does not equal 40, this is not correct either.
We know the sum will only get bigger if H=20, so that is not an option we need to check.
Trying S=7 so H - CS = 1 If H=12 then CS=11. We know that F=8, E=1, and B=1. We are assuming S = 7. The sum of all the amounts is 40. S x F = 7 x 8 = 56. Since 40 does not equal 56, this combination is not correct.
If H=16 then CS=15. The sum of all the amounts is 48. Since 48 does not equal 56, this is not correct either.
Finally, if H=20 then CS=19. The sum of all the amounts is 56. This is equal to S x F, so we have solved it!
H=20, CS=19, S=7, F=8, E=1, and B=1
The Kitchen Clue #7
Change each number into a letter to reveal a six-letter word. A common way to go from numbers to letters would be 1=A, 2=B, 3=C, etc.
The Kitchen Solution
The amounts are 1, 7, 8, 1, 19, 20. Changing those to letters (where A=1 and Z=26) yields the answer AGHAST.
Phew! Love it or hate it, congratulations on getting here.