Step 1: You are given information about DOUGH.

Step 2: TOMATO SAUCE and TOMATO SLICES are both tomato products, so they are on the pizza by the DOUGH clue.

Step 3: Since the TOMATO SAUCE clue says that three ingredients from each row and column are on the pizza, and DOUGH and TOMATO SAUCE are two of the three for the first column, there is only one left. The TOMATO SLICES clue says that the third is either ricotta or mozzarella, so PESTO cannot be on the pizza.

Step 4: The PESTO clue says that any ingredient related to cauliflower is on the pizza, so BROCCOLI is on the pizza.

Step 5: The BROCCOLI clue says that two cheeses are on the pizza. We know from the TOMATO SLICES clue that one is either mozzarella or ricotta, but not both. The only other cheese is PARMESAN, so it must be the other cheese on the pizza.

Step 6: The PARMESAN clue says that the pizza lacks fungi, so there are no MUSHROOMS on the pizza.

Step 7: The MUSHROOM clue says that three ingredients beginning with a vowel are on the pizza. There are four ingredients beginning with a vowel: ONIONS, OLIVES, EGGPLANT, and OREGANO. Since there are already two ingredients in the third column on the pizza, only one more ingredient from that column is on the pizza (i.e. OLIVES and ONIONS cannot both be on the pizza). Therefore, in order to have three ingredients beginning with a vowel on the pizza, both EGGPLANT and OREGANO must be on the pizza.

Step 8: Since BROCCOLI, EGGPLANT, and OREGANO are known to be on the pizza, no more ingredients from the fifth row can be on the pizza due to the TOMATO SAUCE clue. Therefore, neither RICOTTA nor BACON is on the pizza.

Step 9: Since RICOTTA is not on the pizza, then MOZZARELLA must be on the pizza because of the TOMATO SLICES clue. In addition, there are now three ingredients from the first column on the pizza, which satisfies the TOMATO SAUCE clue.

Step 10: The MOZZARELLA clue states that half (two of the four) of the ingredients ending in the letter “I” are on the pizza. The four are SALAMI, PEPPERONI, BROCCOLI, and ZUCCHINI. We know BROCCOLI is one of the two that is on the pizza, so only one of the other three ingredients is on the pizza (i.e. SALAMI and PEPPERONI cannot both be on the pizza). Since BACON is not on the pizza, only one other ingredient from the meat column is not on the pizza; therefore, SALAMI and PEPPERONI cannot both be omitted from the pizza. In other words, one of the two is on the pizza, and the other is not. Since BROCCOLI and either SALAMI or PEPPERONI must be on the pizza, we know ZUCCHINI and the other meat ending in “I” are not on the pizza.

Step 11: In order to have three ingredients in the fourth row be on the pizza, we know SAUSAGE must be on the pizza.

Step 12: Since BACON is not on the pizza and we know that either SALAMI or PEPPERONI is the other ingredient from the second column not on the pizza, then HAMBURGER must be on the pizza in order to ensure three ingredients will be on the pizza from that column.

Step 13: The OREGANO clue says that if HAMBURGER is on the pizza, then OLIVES are not. Since we know that HAMBURGER is on the pizza, the clause about the OLIVES is true. OLIVES are not on the pizza.

Step 14: Since MUSHROOMS and OLIVES are not on the pizza, then ONIONS must be on the pizza in order to have three ingredients from the third column on the pizza. In addition, we now have ONIONS, EGGPLANT, and OREGANO on the pizza, which satisfies the MUSHROOM clue.

Step 15: The RICOTTA clue says that if RED PEPPERS are not on the pizza, then neither is SALAMI. The ONIONS clue says that if RED PEPPERS are on the pizza, then SALAMI is not. We know that RED PEPPERS must either be on the pizza or not be on the pizza, so one of those two statements is true. It does not matter which is true, though, since SALAMI is not on the pizza in both situations.

Step 16: In order to have three ingredients from the second column on the pizza, PEPPERONI must be on the pizza. In addition, we now have two of the four ingredients ending in an “I” (PEPPERONI and BROCCOLI) on the pizza to satisfy MOZZARELLA’s clue.

Step 17: Since PEPPERONI and PARMESAN are known to be on the pizza, then PESTO (already known) and PARSLEY must be the two “P” ingredients not on the pizza in order to satisfy BACON’s clue.

Step 18: In order to have three ingredients from the third row on the pizza, RED PEPPERS must be on the pizza.

Step 19: HAMBURGER’s clue states that if BASIL is not on the pizza, then neither is PEPPERONI. We know PEPPERONI is on the pizza, therefore, BASIL must be on the pizza in order to ensure that HAMBURGER’s statement is correct.

Step 20: There are now three ingredients from both the second row and fifth column on the pizza. Therefore, neither SPINACH nor CHIVES is on the pizza.

Step 21: In order to have three ingredients be present from the first row and fourth column, GREEN PEPPERS must be on the pizza.