Miscellaneous Problems

Problem 10. Brian tore out several successive pages from a book. The first page that he tore up was page number 143. The last page that he tore up is also a three-digit number written with the same digits {1,4,3} but in a different order. How many pages did he tear up?

Problem 9. All the natural numbers starting with 1 are listed consecutively: 1234567891011121314151617181920212223 . . . Which digit occupies the 1002nd place?

Problem 8. Doing only one multiplication, prove that (666)(222) + (1)(333) + (333)(222) + (666)(333) + (1)(445) + (333)(333) + (666)(445) + (333)(445) + (1)(222) = 1000000.

Problem 7. Each element of the set {10, 11, 12, . . . ,19, 20} is multiplied by each element of the set {21, 22, 23, . . . ,29, 30}. If all these products are added, what is the resulting sum?

Problem 6. A certain calculator gives as the result of the product 987654·745321 the number 7.36119E11, which means 736,119,000,000. Explain how to find the last six missing digits.