Frog and lily pad puzzle.

This week we will have only a logic/math puzzle. I’ll break it down into four versions. In all versions there are a specified number of lily pads in a row and a frog who will always jump to an adjacent lily pad every night. Every day you may check just one lily pad for the frog.

Question 1

There are three lily pads. How can you catch the frog in two days?

Question 2

There are five lily pads. How can you catch the frog in six days?

Question 3

There are n lily pads, where n is an odd number. How can you catch the frog in 2n-4 days?

Question 4

There are n lily pads, where n is an even number. How can you catch the frog in 2n-3 days?

Answer 1

Check the middle lily pad the first day. If he isn’t there, check the middle one again the next day.

Answer 2

Number the lily pads as shown in the question.

Any of the following solutions will work.

1. 2,3,4,2,3,4
2. 2,3,4,4,3,2
3. 4,3,2,2,3,4
4. 4,3,2,4,3,2

 

Note that all of them have two groups of 2-3-4, in either ascending or descending order.

Answer 3

1. Start with lily pad 2 and move one lily pad higher until you get to n-1.
2. Repeat step 1

 

You may go in either direction (low to high or high to low) in both steps 1 and 2.

Answer 4

1. Start with lily pad 2 and move one lily pad higher until you get to n-1.
2. Guess any odd numbered lily pad.
3. Repeat step 1

 

You may go in either direction (low to high or high to low) in both steps 1 and 3.

Solution 1

If the frog wasn’t on the middle lily pad the first day, he must jump to it that night. Check it again the next day and you’ll find him for sure.

Solution 2

Start by assuming the frog is on an even-numbered lily pad.

1. Day 1: Pick lily pad 2.
2. If he wasn’t on 2, then he must have been on 4. In that case, he will jump to 3 or 5 the next day.
3. Day 2: Pick lily pad 3.
4. If he wasn’t on 3, then he must have been on 5. In that case, he will jump to 4 the next day.
5. Day 3: Pick lily pad 4.
6. If he wasn’t on 4, then our assumption he started on an even number must have been wrong, in which case he started on an odd number. After three days, he will be on an even number on day 4.
7. Go back to step 1. We know for sure he is on an even number, so we will get him by the same logic within three more guesses.

 

Solution 3

Follow the same logic as solution 2, advancing one lily pad at a time. If he started on an even number, you’ll get him by the time you get to n-1. Otherwise, your assumption he started on an even number was wrong. You’ve already used n-2 guesses, which must be an odd number, so he must be on an even number now. Then go through the same process again, starting at 2 and going to n-1.

Solution 4

Follow the same logic as solution 3, advancing one lily pad at a time. If he started on an even number, you’ll get him by the time you get to n-1. Otherwise, your assumption he started on an even number was wrong. You’ve already used n-2 guesses, which must be an even number, so he must still be on an odd number. Guess any odd number you wish. If that wild guess doesn’t work, he must be on an even number the next day. Then go through the same process again, starting at 2 and going to n-1.