Circle 2 Problem 6 Palindrome Worksheet With Answers

Circle 2 Problem 6 Palindrome Worksheet With Answers - 2011-2012 Page 5

Number Sense

2011/2012 Circle 2 Problem 6

b) The completed hundred chart is shown below.

Observed patterns:

• The portion of the chart from 11 to 99, excluding multiples of 10, displays complete symmetry

about the diagonal zero-step palindromes 11, 22, ..., 99.

• Most of the numbers are one-step palindromes.

• There are no two-step palindromes above the diagonal formed by 19, 28, 37, ..., 91.

• The three- and four-step palindromes occur for numbers greater than 58.

Some further thought may lead students to see why the above types of palindromes occur where

they do in the chart.

• One-step palindromes occur if the sum, S, of the digits of the number is less than 10, or

equal to 11, and the palindrome is SS if S < 10 (e.g., if the sum is 7 then the palindrome is

77), or 121 if S = 11.

• Two-step palindromes occur if S = 10, 12, or 13; three-step palindrome occur if S = 14, and

four-step if S = 15.

Extension:

1. a) The numbers 77, 88, 99 are zero-step palindromes. The numbers 8, 9, 17, 18, 27, 29, 38,

47, predict that 80, 90, 71, 81, 72, 92, 83, 74 will be one-step palindromes, as is the number

100. The numbers 19, 28, 37, 39, 48, 49, 57, 58, 67 predict that 91, 82, 73, 93, 84, 94, 75,

85, 76 will be two-step palidromes. The numbers 59, 68 predict 95, 86 will be three step,

and 69 predicts 96 will be four-step.

b) The remaining six numbers are pairs 78 and 87, 79 and 97, and 89 and 98.

Circle 2 Problem 6 Palindrome Worksheet With Answers - 2011-2012 Page 5