Miscellaneous Problems

Problem 15. A frog is in a 10 ft well. At the beginning of each day, it leaps 5 ft up, but at the end of the day it slides 4 ft down. After how many days, if at all, will the frog escape the well?

Problem 14. Judith was imprisoned by a band of mathematicians and sent to Guantánamo for crimes against Mathematics. Through the mercy of the Brahmin mathematician, she was given the choice of being released after 10 years or be given freedom if she climbed the 100 steps of a 100-step staircase subject to the following
rules:

• She climbs up or down only one step per day.
• She climbs up on every day of January, March, May, July, September, and November.
• She goes down on every day of February, April, June, August, October, and December

Being adept at climbing, she chose this later option. If Judith started on January 1 2001, when will she gain her freedom?

Problem 13. Using all the digits {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, form two 5-digit numbers so that their difference is as large as possible.

Problem 12. Using all the digits {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, form two 5-digit numbers so that their difference is as small as possible.

Problem 11. You and I play the following game. I tell you to write down three 2-digit integers between 10 and 89. Then I write down three 2-digit integers of my choice. The answer comes to 297, no matter which three integers you choose (my choice always depends on yours). For example, suppose you choose 12, 23, 48. Then I choose 87, 76, 51. You add 12 + 23 + 48 + 87 + 76 + 51 = 297. Again, suppose you chose 33, 56, 89. I then choose 66, 43, Observe that 33 + 56 + 89 + 66 + 43 + 10 = 297.

Explain how I choose my numbers so that the answer always comes up to be 297!