# Which step is creaky?

The problem is as follows:

Ms. Flor has a 50-step wooden staircase that allows access from the first floor to the second floor. One of the steps creaks when a person stands on it. Diego and Mike are going to climb the first 15 steps. Diego stood on the first step and then climbed to the end jumping two by two. Mike from the floor jumped three steps and back one step, then jumped three steps and back one and so on to the end. In no case was the creaking step heard. What should be the step that creaks in these first 15 ?

The choices given in my book are as follows:

1. 2nd step
2. 10th step
3. 14th step
4. 6th step
5. none of them

I found this question in my puzzles book Reason and Logic from 2000s. This problem seems to be an adaptation from a reprinted copy of the Martin Gardner's Puzzle Carnival of the 70s.

Regarding on how to approach this problem is the part where I'm still stuck, as I'm not sure what jumping "two by two" and "jumping three and back one" means.

I've illustrated my question in the figure below: As it can be seen in the first sketch, Diego starts at the first step and then jumps two by two, leaving two steps untouched between. Then Mike follows him from the floor by jumping three, in other words moving straight to the 4th and then returning to the 3rd.

In this scenario, the only steps left untouched are the 2nd and the next three. These are, 5, 8, 11 and 14. Thus any of those can be the step being asked.

My other interpretation is the second row of images. Diego stands on the first step and then jumps not to the 4th but instead to the 3rd step, and he goes on and on, always landing in the odd steps.

Mike follows him but instead of jumping forward three and back one, and landing on the 4th step as indicated in the first picture, he would land on the 3rd step and then back to the 2nd step. Then he would repeat this sequence, and along with Diego, no step is left untouched and since no creak was heard then it meant that such step is not among those 15 steps.

But this is not the answer according to my book. What went wrong in my analysis? Please include a diagram in your answer, but more importantly, a detailed step by step (no pun intended) solution. There's one more thing which I'm curious. Could had this been modeled by a series?

Initially I thought it would follow:

$$t_n=a_{1}+(n-1)3=1+(n-1)3=3n-2$$

for Diego. But for Mike it would be different and I don't think it could be modeled using an arithmetic series. Would this approach be any help? The problem is only about 15 steps, and making an extended drawing for all 50 steps seems impractical.

• I think your error is "Diego starts at the first step and then jumps two by two, leaving two steps untouched between" - if someone were to jump one by one, there would be no steps untouched in between, right? So two by two is every other, leaving one untouched in between. Try that. Nov 6 '20 at 7:38
• @KateGregory Upon looking at my method it seems that my drawing was not right I mean the first interpretation, the second interpretation seems to be it. But it is a little bit weird since jumping three by three, wouldn't it mean leaving a three steps untouched first?. Nov 8 '20 at 2:38

Your second interpretation is the correct one, but your picture of the steps that Mike visits is incorrect.

Diego visits steps

1, 3, 5, 7, 9, 11, 13, 15

Mike visits steps:

3, 2, 5, 4, 7, 6, 9, 8, 11, 10, 13, 12, 15

Note that Mike visits

every step except for 1 and 14. He would visit 14 if he were to continue on, but they stop when they reach step 15.

• Thanks for pointing out my error. I double checked my drawing, as you mentioned I totally overlooked the fact that he wouldn't back one step after landing on $15$, since Diego visited $1$ this would meant that $14$ is untouched. My book says the answer is $6$ but there has to be an error or typo, your answer is the most logical one. Nov 8 '20 at 2:24