Ageless Advice



1. The ages are 1, 5, and 7.

2. The ages are 2, 3, and 6.

3. The ages are 1, 6, and 6. To get the answer, first look at all the combinations of three ages that when multiplied yield 36, with the corresponding sums of those ages:



Note that there are two combinations of ages that add up to 13. No other sums appear twice in this table, so the sum on the sign must have been 13. There is a set of twins in each combination of ages, but from the hint we know that only the youngest has red hair. If one child is the youngest, she or he can't be a twin, so the answer must be 1, 6, and 6.

4. Using the same logic as in problem 3, you'll find that the only ambiguous sum for which there are two age combinations is 14 (2 + 6 + 6 and 3 + 3 + 8). The solution is 2, 6, and 6—the set of ages in which the youngest isn't a twin.

5. The next number that could work is 225, which yields the ambiguous sum 31 (1 + 15 + 15 and 3 + 3 + 25). The next number after that is 288. In this instance, there are two different possible ambiguous sums: 26 (2 + 12 + 12 and 4 + 4 + 18) and 42 (1 + 9 + 32 and 2 + 4 + 36). It was the fact that the youngest is not a twin that allowed the dad to solve the puzzle, so the ambiguous sum on the sign must have been 26, and the combination of ages was 2, 12, and 12.



Tomb Etchings


1.



2.



3.





Do you know your ABCs?



1. The arrangement of letters obeys rules c, e, and f.

2.



3. The redundant one is rule f, which is a consequence of rule b. The only way to avoid A's being adjacent to B's (rule b) is to put all the A's to one side of the grid and "buffer" the A's with C's, in which case two of the A's must always be adjacent to the third A (rule f).


4. Rules d and e. Without rule d you can reflect the solution about a vertical line—that is, you can transpose the left-hand and right-hand columns for an alternate solution, as shown at top right. Without rule e you can reflect the solution about a diagonal line as shown bottom right.

5. The answer is rule b. If you're wondering how to determine the number of legal arrangements that each rule by itself dictates, here's a quick rundown. The rules are listed in order from the highest to the lowest number of legal arrangements.

Rule d: 120,960 arrangements, which is the total number of arrangements divided by 3. To count the total number of ways to arrange the grid, consider the nine squares of the grid one at a time and the fact that you have only nine different letters to place in those squares. We have nine choices for the letter in square 1. For each of these nine choices, there are eight choices for square 2 (we used up one of our nine letters in the first square), for a total of 72 (9 x 8) choices. Then for each of those 72 choices for squares 1 and 2, there are seven choices for square 3, yielding 504 possible choices (9 x 8 x 7). Continue with that reasoning, and the total number of choices is 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 362,880, or 9 factorial (also written as 9!).

Rule e: Also 120,960 arrangements, which is the total number of arrangements divided by 3.

Rule f: 95,040 arrangements, which is the number of ways to place three A's (22) times the number of ways to color the three A's (3!, or 6) times the number of ways to arrange the remaining 6 letters (6!, or 720), or 22 x 6 x 720.

Here are the 22 ways to arrange the three A's:




Rule a: 7,776 arrangements, which is the number of possible color patterns (6) times the number of colorings for each arrangement (3!, or 6) times the number of arrangements of letters within each coloring (3! x 3! x 3!, or 6 x 6 x 6, or 216).

Here are the six basic patterns for colors:




Rule c: 2,880 arrangements, which is the number of arrangements of the five letters that are either the letter C or yellow or both (5!, or 120) times the number of arrangements of the remaining four letters (4!, or 24). The key is to recognize that there is only one way to arrange the five letters that are either C or yellow so that no two are next to each other, as shown below.




Rule b: 1,728 arrangements, which is the number of possible arrangements of the letters (8) times the number of colorings (3! x 3! x 3!, or 6 x 6 x 6, or 216).

Here are the eight letter patterns that prevent A and B from being neighbors:







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