## Armstrong Numbers

• Lesson
6-8,9-12
2

An Armstrong number is an n-digit number that is equal to the sum of the nth powers of its digits. In this lesson, students will explore Armstrong numbers, identify all Armstrong numbers less than 1000, and investigate a recursive sequence that uses a similar process. Throughout the lesson, students will use spreadsheets or other technology.

Explain to students that they will be implementing the following process for several numbers:

1. Raise each digit to a power equal to the number of digits in the number. For instance, each digit of a four‑digit number would be raised to the fourth power; each digit of a five‑digit number would be raised to the fifth power; and so on.

Demonstrate this process for 123—because 123 is a three-digit number, raise each digit to the third power, and add: 13 + 23 + 33 = 36. If necessary, show other examples. Then, have students implement the process for the following numbers:

 6 23 99 231 1634 11,111

Students are likely to notice that two numbers from this list—6 and 1634—return their original value when this process is applied; that is, 61 = 6, and 14 + 64 + 34 + 44 = 1634.

Inform students that numbers which return their original value are known as Armstrong numbers. Once you are certain that all students understand this idea, allow them to work in pairs to create a succinct definition of Armstrong numbers. After a few minutes, have each group share their definition with another pair, and then allow the class to vote on which definition is most lucid and precise. For comparison, share the following formal definition:

Armstrong number: An n-digit number equal to the sum of the nth powers of its digits.

Finding Armstrong Numbers

In examining the numbers above, students found that 6 and 1634 were Armstrong numbers. Now ask them to determine all Armstrong numbers less than 1000.

There are several strategies that students could employ to solve this problem. Allow students some time to come up with a strategy on their own. If students are stuck, suggest one of the following:

• Using a list of numbers 1‑999, students can eliminate any non‑Armstrong numbers. (Many numbers can be eliminated from this list by inspection. For instance, any number ending in two 0’s, such as 100, 200, 300, …, can be eliminated, because the cube of the hundreds digit will not end in two 0’s.) While this method can produce valid results, it is susceptible to human error, and it may take a long time.
• Using a programmable calculator or a computer, write a program to generate and test all numbers 1‑999. (Such a program could generate the numbers by repeating a loop of the form "For n = 0 to 9" three times.)
• Using a spreadsheet, parse each number 1‑999 into its three digits, and then calculate the sum of the powers. The spreadsheet Armstrong-Students.xls can be given to kids as a starting point for their investigation; students will need to enter formulas to find the units digit in column D and the sum of the cubes in column E. Alternatively, you may choose to have students begin with a blank spreadsheet and discern a formula for parsing the hundreds and tens digits, too. The file Armstrong-Teacher.xls shows one possibility for creating formulas and testing numbers.

As it turns out, there are nine single-digit Armstrong numbers, namely 1‑9; there are no two-digit Armstrong numbers; and there are four three-digit Armstrong numbers: 153, 370, 371, and 407.

Strong Arm Iteration

The investigation of three-digit Armstrong numbers is conducted by adding the cubes of the digits. This process of cubing the digits can be continually repeated on the result to reveal some interesting patterns.

The process of Strong Arm Iteration works as follows:

1. Begin with an n-digit number.
2. Raise each digit to the nth power, and compute the sum.
3. Raise each digit of the result to the nth power, and compute the sum.
4. Repeat Step 3 until a pattern emerges.

For instance, take the number 123. Cube its digits and add them: 13 + 23 + 33 = 36. Now, take the digits of the result, cube them, and add: 33 + 63 = 243. Continue by cubing the digits and adding for each result:

243: 23 + 43 + 33 = 99
99: 93 + 93 = 1458
1458: 13 + 43 + 53 + 83 = 702
702: 73 + 03 + 23 = 351
351: 33 + 53 + 13 = 153
153: 13 + 53 + 33 = 153

Notice that the process eventually leads to 153, which gives itself when the process is continued.

Allow students to explore other three-digit numbers. As before, you may ask them to construct a computer program or spreadsheet on their own to investigate this problem; or, you might supply them with the file Armstrong-Iteration.xls to use for investigation.

To guide this investigation, you may wish to distribute the Strong Arm activity sheet to your students.

 Strong Arm Activity Sheet

In particular, ask them to consider the following questions:

1. If you begin with 123, the sequence reaches 153, and then it begins to repeat. That is, 13 + 53 + 33 = 153. What other three‑digit numbers will eventually reach 153 and begin to repeat? Is there a pattern to the numbers that reach 153?
[There are many numbers that reach 153 and then repeat; some of the numbers are 135, 213, 369, 423, 546, 678, 775, 819, and 972. In fact, all multiples of 3 eventually reach 153.]

2. Other than 153, what other numbers are reached when this process is applied?
[The other three‑digit Armstong numbers are occasionally reached. For instance, 124 eventually leads to 370; 551 leads to 371; and 740 leads to 407.]

3. What other interesting things did you notice during this investigation? If possible, explain why these interesting things happened.
[Several numbers lead to a cycle of repetition rather than to an Armstrong number. For instance, 136 leads to 244 which leads back to 136, and this two‑number cycle repeats. Cycles of three numbers also occur.]

References

• Knuth, Donald E. "Are Toy Problems Useful?" reprinted in "Selected Papers on Computer Science," Stanford, CA: Center for the Study of Language and Information, Leland Stanford Junior University, 1996.
• L. Deimel, Jr., and M. Jones. "Finding Pluperfect Digital Invariants." Journal of Recreational Mathematics, Vol. 14:2, 1981‑82, p. 87‑107.

Assessments

1. Require students to write a journal entry that:
1. identifies all of the Armstrong numbers less than 1000, and
2. explains how they know that they’ve found all of them.

2. Review students’ spreadsheets or computer programs to determine their method for finding Armstrong numbers. Even if students were not able to find all Armstrong numbers less than 1000, rate their work based on the approach they used.

Extensions

1. In total, there are 88 Armstrong numbers. Find them all.

The Delphi For Fun web site poses the following problem about Armstrong numbers:

The front of a shirt says, "The smallest three‑digit number equal to the sum of its digits cubed." What number should be on the back of the shirt?

Gary Darby of Delphi For Fun generously provided Illuminations with a special version of the T-Shirt Program, which can be used to find Armstrong numbers. (This program is able to identify all four of the three‑digit Armstrong Numbers in less than one‑tenth of a second. Given enough time, it can also generate a full list of all 88 Armstrong Numbers.)

2. The number 115,132,219,018,763,992,565,095,597,973,971,522,401, which has 39 digits, is the largest Armstrong number. Explain why there can be no Armstrong numbers with more digits.
3. The number 153 is a special number:
• It is an Armstrong number: 13 + 53 + 33 = 153.
• It is a triangular number: 1 + 2 + 3 + … + 17 = 153.
• It is a sum of factorials: 1! + 2! + 3! + 4! + 5! = 153.

Have students investigate other mathematical and real-world occurrences of the number 153.

Questions for Students

1. There are 27 three‑digit numbers such that each digit is greater than 6. Without checking each possibility, how can you be sure that none of these numbers are Armstrong numbers?
[The sum of the cubes of the digits would be at least 73 + 73 + 73 = 1049, which is greater than any three‑digit number.]
2. How do you know that you’ve found all of the Armstrong numbers less than 1000?
[A systematic process should be employed. Possible processes include using a spreadsheet to check each number 1‑999, or logically removing numbers that are obviously not Armstrong Numbers and then checking those that remain.]
3. What patterns do you notice when the Strong Arm Iteration process is used on three‑digit numbers?
[The most interesting pattern is that all three‑digit multiples of 3 eventually reach 153. Other numbers enter a two‑number cycle, while others enter a three‑number cycle.]

Teacher Reflection

• Were students able to handle the mathematical content contained in the lesson? If not, what could be done to make it more understandable?
• How were you able to challenge the high achievers? How did you extend the investigation to keep them interested?
• Did you find it necessary to make adjustments while teaching the lesson? If so, what adjustments, and were these adjustments effective?

### Learning Objectives

Students will:

• Formulate a definition of Armstrong numbers based on several examples
• Identify all Armstrong numbers less than 1000 using appropriate technology
• Reasonably explain why the sequence of Armstrong numbers is finite

### NCTM Standards and Expectations

• Develop an understanding of large numbers and recognize and appropriately use exponential, scientific, and calculator notation.
• Use number-theory arguments to justify relationships involving whole numbers.
• Judge the effects of such operations as multiplication, division, and computing powers and roots on the magnitudes of quantities.
• Develop fluency in operations with real numbers, vectors, and matrices, using mental computation or paper-and-pencil calculations for simple cases and technology for more-complicated cases.