## 5th and 6th Grade, December, 2017

B.1161. We glued 8 identical regular dice into a 2x2x2 cube. What is the least number of dots on the outside of the big cube if the sides of each of the dice have 1 to 6 dots on them so that any two facing sides have a total of 7 dots on them? (Each of the dice has 6 different numbers of dots on its sides.)

B.1162. Lassie decided to go home. It took him 4 days: the first day he made 5/8 of the whole distance, the second day he made 1/12 of the whole distance, on the third day he ran 5 km, and for the fourth day he still had 1/4 of the whole distance. How far was he from home?

B.1163. The station manager at the Hoboken train station puts up a Christmas tree every year. He has 5 different color light bulbs which can be turned on and off independently from one another. The manager puts a few light bulbs on the tree on December 23rd. How many different ways can he pick those light bulbs if he wants them to light up in a different color combination every day till January 6th?

B.1164. One of the famous Hungarian mathematicians lived all his life in the 19th century. Three of the digits in the years of his birth and his death are the same. His birth year is a multiple of 17, and the year of his death is a multiple of 31. What year was he born and what year did he die if you know that he lived for more than 50 years? (You can get an extra point for his name.)

B.1165. The currency in Payland is called a hollar. The mass of four 20-hollar coins is the same as the mass of five 10-hollar coins. If 1 kg of 10-hollar coins is worth 5000 hollars then what are 3 kg of 20-hollar coins worth?

B.1166. 25 of the passengers on a bus know how to ride a bicycle, 20 of them know how to swim, and there is no person on the bus who would not know how to do either. The sum of the digits of 6 times the number of passengers is 3 times as much as the sum of the digits of the number of passengers. How many passengers can be on this bus? How many of them know how to ride a bike but do not know how to swim, and how many of them know how to swim but do not know how to ride a bike?

B.1167. What positive whole number could “a” be if 2/7 < 3/a < 4/9?

B.1168. Tom spends 1/4 of his weekly allowance on Monday, 1/6 of it on Tuesday, 1/8 of it on Wednesday, 1/12 of it on Thursday, 1/24 of it on Friday. This means an average spending of \$1.60 a day for those days. How much money does Tom have left for the weekend?