快猫短视频

#123 Cara鈥檚 fleet of cars

Cara is a car enthusiast. In autumn 2001, she was lucky enough to inherit a fortune from her great-aunt Mildred, and decided to create her own fleet of cars by buying a new one twice a year for the next 10 years.

UK car number plates include a two-digit number to indicate the year of registration, with two of these numbers issued each year. For example, cars registered in 2004 have 04 or 54 on the plate. The cars Cara bought in the autumn had number plates with 51, 52, 53 and so on, up to 60. The cars she bought in the spring had number plates with 02, 03, 04 and so on, up to 11.

Cara happens to love number patterns and wanted to park all her cars in a row so that in any pair of adjacent vehicles, one plate was a multiple of the other. For example, 56 could be parked next to 08 because 56 is a multiple of 8. No matter how she tried, she couldn鈥檛 make a row using all the cars. What was the longest row that Cara could make?

• Answer next week

#122 Article of faith

Solution

Dan finishes on page 20. Since he halves his counting three times, the number of pages he first counts forward must be divisible by 8, call it 8k.

So overall he counts forward 8k+2k (=10k) and counts back 4k+k (=5k), and the net result is that he counts forward 5k. Call the page he starts on N. We are told that 5k+N = 4N, so 5k=3N, which means k is a multiple of 3.

And since there are fewer than 50 pages in the magazine, only k=3 works. So he starts on page 5 and counts on 24 from there, then back 12, forward 6 and back 3 to end on page 20.