**Puzzle 61**

One morning, Freddy Frog wakes up at dawn and, not remembering that he fell asleep on the edge of a 30-foot-deep well, makes one hop in the wrong direction and falls to the bottom of the well. He immediately starts clawing his way up the wall of the well and by sunset at the end of his first day in the well, he manages to climb up three feet. Unfortunately, during the nighttime, he falls asleep, his grip loosens and he slides down two feet. From dawn until sunset on the second day, Freddy crawls up three feet and during the second night, he slides back two feet. Assuming that this pattern of progress continues on every successive day and night, how many *days* does it take him to crawl his way out of the well?

*out*of the well. So we need to ask ourselves: On which day does Freddy reach the top of the well? At the end of 27 days and 27 nights, Freddy is 27 feet from the top of the well. On the 28th day, Freddy climbs up three feet and reaches the top of the well and hops away. Answer: 28 days. The general solution, independent of the depth of the well, can be found by solving simpler analogous puzzles. If the well were three feet deep, the frog would reach the top of the well in one day. If it were four feet deep, at the end of one day and one night, he would reach a point that is three feet from the top of the well, so he’d reach the top of the well at the end of the second day. By the same token, he’d reach the top of a five-foot-deep well in three days, and so on. Therefore, for

*n*≥ 3, it takes him

*n*– 2 days for him to get out of an

*n-*foot-deep well.

**Puzzle 62**

Find *x*, *y* and *z* if = .

*x*,

*y*and

*z*is the sum of an integer and a complex fraction, it occurs to us that if we change into a complex fraction, we may obtain an expression that has the same structure as the expression containing

*x*,

*y*and

*z*. Hence, , and comparing the first and last expressions, we see that

*,*and.

**Puzzle 63**

Joe Hill owns an isosceles-right-triangular plot of land with each of the two congruent sides measuring one mile. One day he starts walking from the vertex of this triangle to the midpoint of its hypotenuse, his path dividing the triangle into two smaller isosceles right triangles. He repeats his path, using one of those smaller triangles, thereby dividing *that* triangle into two even smaller isosceles right triangles. Not having much else to do, he continues to repeat his path with smaller and smaller triangles. Assuming he can take infinitesimal steps in infinitesimal time periods (which could only happen in a math puzzle), at the end of his trek, how far has he walked?

*r*is the common ratio of each term to its predecessor, the sums of these two sequences are and 1, respectively, so Joe’s entire walk measures miles.

**Puzzle 64**

Find a positive integer whose third and fourth powers together contain all ten digits from 0 to 9 exactly once.

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**Puzzle 65**

Arrange six toothpicks to form four equilateral triangles without bending or breaking any of the toothpicks.

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