Square Fun With Blocks

Number Theory Level 2

You have a set of blocks in school which your math prof has numbered in chalk from 1 to 100. Your math prof says,

"Remove all the blocks that are numbered with a perfect square, and then renumber the remaining blocks consecutively starting from 1. Keep repeating that till you have removed all the blocks."

Can you tell your math prof quickly how many times the operation will need to be performed to reduce the number of blocks in the set to zero ?


The answer is 19.

Satyen Nabar by Satyen Nabar , 48, India

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3 solutions

Mohd Faraz
Apr 20, 2014

10,9 9,8 8,7 7,.....;therefore 2*9+1=19

3 Helpful 0 Interesting 0 Brilliant 0 Confused
Anzar Aznzar
Mar 30, 2014

Perfect Squares are : 1 , 4 , 9 , 16 , 25 , 36 , 49 , 64 , 81 and 100

1) 100 - 10 = 90

2) 90 - 9 = 81

3) 81 - 9 = 72

4) 72 - 8 = 64

5) 64 - 8 = 56

6) 56 - 7 = 49

7) 49 - 7 = 42

8) 42 - 6 = 36

9)36 - 6 = 30

10) 30 - 5 = 25

11) 25 - 5 = 20

12) 20 - 4 = 16

13) 16 - 4 = 12

14) 12 - 3 = 9

15) 9 - 3 = 6

16) 6 - 2 = 4

17)4 - 2 = 2

18) 2 - 1 = 1

19) 1 - 1 = 0

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Anuj Mittal
Mar 29, 2014

1). Between numbers 1 to 100, there are perfect squares of numbers 1 to 10. Therefore, 10 blocks are removed. Remains - 90

2). Between numbers 1 to 90, there are perfect squares of numbers 1 to 9. Therefore, 9 blocks are removed. Remains - 81

3). Between numbers 1 to 81, there are perfect squares of numbers 1 to 9. Therefore, 10 blocks are removed. Remains - 72

4). Between numbers 1 to 72, there are perfect squares of numbers 1 to 8. Therefore, 10 blocks are removed. Remains - 64

5). Between numbers 1 to 64, there are perfect squares of numbers 1 to 8. Therefore, 10 blocks are removed. Remains - 56

6). Between numbers 1 to 56, there are perfect squares of numbers 1 to 7. Therefore, 10 blocks are removed. Remains - 49

7). Between numbers 1 to 49, there are perfect squares of numbers 1 to 7. Therefore, 10 blocks are removed. Remains - 42

8). Between numbers 1 to 42, there are perfect squares of numbers 1 to 6. Therefore, 10 blocks are removed. Remains - 36

9). Between numbers 1 to 36, there are perfect squares of numbers 1 to 6. Therefore, 10 blocks are removed. Remains - 30

10). Between numbers 1 to 30, there are perfect squares of numbers 1 to 5. Therefore, 10 blocks are removed. Remains - 25

11). Between numbers 1 to 25, there are perfect squares of numbers 1 to 5. Therefore, 10 blocks are removed. Remains - 20

12). Between numbers 1 to 20, there are perfect squares of numbers 1 to 4. Therefore, 10 blocks are removed. Remains - 16

13). Between numbers 1 to 16, there are perfect squares of numbers 1 to 4. Therefore, 10 blocks are removed. Remains - 12

14). Between numbers 1 to 12, there are perfect squares of numbers 1 to 3. Therefore, 10 blocks are removed. Remains - 9

15). Between numbers 1 to 9, there are perfect squares of numbers 1 to 3. Therefore, 10 blocks are removed. Remains - 6

16). Between numbers 1 to 6, there are perfect squares of numbers 1 to 2. Therefore, 10 blocks are removed. Remains - 4

17). Between numbers 1 to 100, there are perfect squares of numbers 1 to 2. Therefore, 10 blocks are removed. Remains - 2

18). Between numbers 1 to 2, there are perfect squares of numbers 1 to 1. Therefore, 10 blocks are removed. Remains - 1

19). Between numbers 1 to 1, there are perfect squares of numbers 1 to 1. Therefore, 10 blocks are removed Remains - 0

thus, step must be repeated 19 times

1 Helpful 0 Interesting 0 Brilliant 0 Confused

anuj mittal you explained it perfectly

Siddharth Srivastava - 7 years, 1 month ago

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