Reading by Numbers

This activity provides a further opportunity for students to investigate number patterns. For students who are keen to make sense of how numbers work, investigating palindromic numbers can be quite fascinating. After the students have tackled question 2, you could tell them about the challenge relating to the number 196
mentioned in the Answers.
A useful extension would be to encourage the students to think of other questions about palindromic numbers that they could investigate, for example: Can palindromic numbers be formed from 4-digit numbers? Is there something special about numbers that form one-step palindromic numbers? Does every palindromic number
form another palindromic number if the process is repeated?
Another way of thinking about questions is to consider them as conjectures. For example, the students might conjecture that it is probably possible to form palindromic numbers from 4-digit numbers, or at least some of them. They then set out to investigate whether their conjecture holds and, if so, under what circumstances.
They may begin with a number such as 4 268 and find that after a number of steps they end up with the palindromic number 2 786 872. They need to see that they cannot generalise from this one instance that palindromic numbers can be made from any 4-digit number. More investigation is needed.
The questions lend themselves to using a calculator for the “number-crunching” so that the students can get on with the investigating aspect. Depending on the class, the activities may be done in pairs as this can often generate more interest than working alone. However, some students prefer to work away quietly on their own, so it may be a matter of providing the choice. Whatever they decide, there is likely to be value in the class sharing results because different students will probably investigate different numbers.
The question about decimal numbers is interesting. The answer for this question indicates that provided the position of the decimal point is fixed, some decimal numbers can be turned into palindromic numbers whereas others cannot, even those using the same digits. For instance, 0.95 works (it becomes 11.11),
whereas 9.5 does not (it becomes 111.1), and neither does 0.095 (which becomes 2.992) or 0.0095 (which becomes 0.5995). The latter three look like palindromic numbers, but to be true palindromic numbers, they would need to have the decimal point in the middle in each case. The students might like to try the decimal number 0.1289. It doesn’t form a palindromic number, but it can be turned into a palindromic number by shifting the decimal point. See if the students can discover this number (it is 1.289). They may also like to try the decimal number 95.95, which can become a palindromic number after a few steps (it becomes 391.193).
The students will probably be able to figure out that if the palindrome-lookalike of a decimal number has an odd number of digits, it can never be a true palindromic number. For example, the decimal number 0.3289 becomes 3.4243 after two steps, but this has five digits and could never be a palindromic number. There is no “middle” where the decimal point could be located.

Answers to Activities

1. a. Numbers will vary. Numbers that finish as one-step palindromic numbers include 10–18, 20–27, 29, 30–36, 38, 40–45, and 47. Numbers that finish as two-step palindromic numbers include 19, 28, 37, 39, 46, 48, and 49.
A number that finishes as a four-step palindromic number is 87.
A number that finishes as a six-step palindromic number is 97.
b. It works for some numbers, but only if the decimal point is kept constantly in the same place. For example:

However, in the following example, it doesn’t work even though the digits used are the same:

2. Answers will vary. Some 3-digit numbers do not produce a palindromic number. (A palindromic number for 196 has not been found in over 200 000 reversals.)
3. 89 finishes as a 24-step palindromic number (8813200023188).