Make an organized list

Mina “Let’s keep the varieties in this order: Basil, Bean, Marigold, Sunflower.”

Li Wei “For each pair, we’ll name the earlier variety first. Then a reversed pair won’t get another card.”

CKSprout “What did the system reveal about the first pile?”

Mina “It repeated Basil–Bean. It missed Basil–Marigold, Bean–Marigold and Bean–Sunflower.”

Li Wei “We can still choose each left-hand variety in turn. This time, all three other varieties are allowed on the right.”

Mina “So Basil on the left and Bean on the right stays different from Bean on the left and Basil on the right. The two positions do different jobs.”

Mina “If we change all three counts at once, it will be easy to skip a mix.”

Li Wei “Let’s fix the Basil count first. For each Basil count, increase Bean one at a time. Sunflower gets the packets left.”

Mina “Planting seven happened last. Before that, there were twenty-one seedlings.”

CKSprout “Which moment does that twenty-one describe?”

Li Wei “After the donation. Half remained, so the original delivery had twice as many: forty-two.”

Li Wei “I’ll increase the large-tray count one at a time. Each count leaves a fixed number of seedlings for small trays.”

CKSprout “How will you know where to stop?”

Mina “Both sizes were there. Eight large trays and even one small tray would already hold forty-three. I only need to try one through seven large trays.”

Mina “One large tray leaves thirty-seven seedlings for small trays. Thirty-seven cannot fill whole trays of three.”

CKSprout “What has that trial told you?”

Li Wei “It rules out that large count, not every mix. I’ll continue through seven.”

Mina “Three large trays leave \(42-15=27\) seedlings. Those fill \(27\div3=9\) small trays.”

Li Wei “Six large trays leave \(42-30=12\). Those fill \(12\div3=4\) small trays. We have two matching rows.”

CKSprout “What shows that your list is complete?”

Mina “We checked every allowed large count. Each fixes one small count, so no mix is missed or repeated. Finding the first matching row would have missed the second one.”

CKSprout “What can the clues tell us about which mix actually arrived?”

Li Wei “Both deliveries end with fourteen seedlings. We cannot choose between them without another clue.”

Mina “I’ll put back the eighteen planted seedlings first. That makes one hundred and twenty-six after donation.”

CKSprout “How is that amount connected to the original delivery?”

Li Wei “It is the three quarters left. One quarter is forty-two, so four quarters make one hundred and sixty-eight.”

Li Wei “Both sizes must be there. I’ll start with one large and thirty-nine small, then increase the large count one at a time. The last possible large count is thirty-nine.”

CKSprout “What fixes the small count in each row?”

Mina “There are forty trays altogether. One large leaves thirty-nine small; two large leave thirty-eight small; and so on.”

Mina “The first three totals are one hundred and twenty-two, one hundred and twenty-four, and one hundred and twenty-six. The total rises by two.”

CKSprout “What would show that the change keeps working?”

Li Wei “Each step replaces one tray of three with one tray of five. There are still forty trays, and every replacement adds two seedlings.”

Mina “Our legal rows run from one hundred and twenty-two to one hundred and ninety-eight seedlings, so one hundred and sixty-eight lies between them. The first row has one large tray. We need forty-six more seedlings from there.”

CKSprout “Where does that put you in your order?”

Li Wei “Each step adds two, so it takes twenty-three more steps. I started at one large tray: one plus twenty-three is twenty-four large trays.”