LONG WINTER HOMEWORK 7 LEVEL 2 DUE 1 st class of Spring Semester, 2017. Part I. Reading assignment: The Number Devil by Hans Magnus Enzensberger Read the book. We plan to have a workshop in Spring related to this reading assignment (The details will be explained later). Math Circle has 5 copies of the book that we will gladly lend to you for the winter break. Part II. DUE 1 st class of Spring Semester, 2017. This is a homework to entertain you during the school winter break. Solve as many problems as you want – it is not required to solve all of them. Write your arguments and solutions clearly. Please do not write solutions between the lines of the problems – there may be not enough space. Use a separate sheet of paper. 1. The integers greater than 1 are arranged as shown. 2 3 4 5 9 8 7 6 10 11 12 13 17 16 … 15 … 14 … a) In which column will 100 fall? b) In which column will 500 fall? 2. The sum of all numbers in the 2 X 2 square 1 2 2 3 equals 8=2 x 2 x 2. The sum of all numbers in the 3 X 3 square 1 2 3 2 3 4 3 4 5 equals 27= 3 x 3 x 3
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LONG WINTER HOMEWORK 7 LEVEL 2 DUE 1st class of Spring Semester, 2017. Part I. Reading assignment: The Number Devil by Hans Magnus Enzensberger Read the book. We plan to have a workshop in Spring related to this reading assignment (The details will be explained later). Math Circle has 5 copies of the book that we will gladly lend to you for the winter break. Part II. DUE 1st class of Spring Semester, 2017. This is a homework to entertain you during the school winter break. Solve as many problems as you want – it is not required to solve all of them. Write your arguments and solutions clearly. Please do not write solutions between the lines of the problems – there may be not enough space. Use a separate sheet of paper.
1. The integers greater than 1 are arranged as shown. 2 3 4 5 9 8 7 6 10 11 12 13 17 16
… 15 …
14 …
a) In which column will 100 fall? b) In which column will 500 fall? 2. The sum of all numbers in the 2 X 2 square 1 2 2 3 equals 8=2 x 2 x 2. The sum of all numbers in the 3 X 3 square 1 2 3 2 3 4 3 4 5 equals 27= 3 x 3 x 3
Can you show that the sum of numbers in the 100 X 100 square
3. In three years, Mark is going to be three times my present age. I will then be half as old as he will be. How old am I now?
4. In my pocket I have 12 coins worth $1.10. I have only nickels, dimes and quarters. What are all of the possible coin combinations?
5. Prove that the sum of any three consecutive numbers is always divisible by 3. (For example, (10+11+12=33 is divisible by 3).
6. There are ten baseball teams in a tournament. Each team plays each of the other teams twice. How many games are played altogether?
7. Using the digits 1 through 9 once each, fill in the boxes to make the equations true: � − � = � ×
� ÷ � = � ||
� + � = �
8. Take any number. Add 10, multiply by 2, add 100, divide by 2, and subtract the original number. Prove that the answer will be always the number of minutes in an hour.
9. Four boys played soccer in the backyard and broke the window. They said the following:
Bob: Charlie did it. Charlie: Eric did it. Dave: I didn’t do it Eric: Charlie lied when he said I did it. a) If only one of the statements is true, who was guilty?
10. Today 1 Estonian Kroon (currency, money of the country Estonia) equals about 7 Japanese Yen (currency of the country Japan), and about 41 Chilean Peso (currency of the country Chilie). What is worth more: 100 Chilean Peso or 15 Japanese Yen? 12. Make an addition problem of three-digit numbers and the answer, using each of the digits 1 to 9 only once: ��� + ��� = ��� 13. What is the smallest counting number that is divisible by numbers 2 through 10? 14. How many 1x 1x1 cubes are in the following stack?
