## Task 2 - 210912

Mr. Schulze meets Mr. Lehmann after a long time and invites him to his home. On the way, he tells Mr. Lehmann that he is the father of three children. Mr. Lehmann wants to know how old they are; he is satisfied with age statements in full years.

Mr. Schulze replies: "The product of the three age statements is 72. The sum of the three age statements is my house number. We have just arrived at our house; you can see my number." To this, Mr. Lehmann responds: "But one cannot uniquely determine the three age statements from this information." "That's true," Mr. Schulze agrees, "but sometime between the birth of the second and the third child, this house was built. We worked on the house for one year and one day." "Thank you! Now the age statements can be uniquely determined," Mr. Lehmann concludes the conversation.

Investigate whether there is a combination of three age statements that satisfy all the statements in this conversation! Investigate whether there is only one such combination! If that is the case, determine this!
The statement about the product of the three age specifications is true for the following combinations:

| 1st Child | 72 | 36 | 24 | 18 | 12 | 9 | 18 | 12 | 9 | $\mathbf{6}$ | 8 | 6 |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 2nd Child | 1 | 2 | 3 | 4 | 6 | 8 | 2 | 3 | 4 | $\mathbf{6}$ | 3 | 4 |
| 3rd Child | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | $\mathbf{2}$ | 3 | 3 |

Mr. Lehrmann's statement "From these specifications, the three age specifications cannot be uniquely determined" is true if and only if the house number is 14. The next statement by Mr. Schulte is true if and only if the 3rd child is at least one year younger than the 2nd child.

Therefore, Mr. Lehrmann's final statement is true if and only if the age specifications are 6, 6, and 2.

Thus, it is shown: There is a combination of the three age specifications for which all statements in the conversation are true; there is also only one such combination; it is 6, 6, and 2.