Tag Archives: Digits

Time Is Power!

While taking a coffee break, my secretary, Sue, happened to glance at her digital watch. It showed the following time:

1 : 4 4

“That’s rather curious,” she thought. “If I remove the two dots, I’ll have the number 144, which is the square of 12. I wonder how likely that sort of thing happens during my workday?”

Assuming that Sue works from 7 a.m. until 7 p.m., what is the probability that her watch will show a square number? Express your answer to the nearest hundredth of a percent.

EXTRA: what is the probability that a cube, 4th power, 5th power, etc. will occur in addition to squares?

SUPER-EXTRA: what is the probability for a power of any kind appearing on a watch set to a 24-hour day? (This means, times can range from 0:00 to 23:59. And “first” powers are not allowed throughout this problem.)

HYPER-EXTRA: if we now use the “seconds” digits that appear on many watches, what is the largest square number that can occur? Largest cube?


No, that is not someone’s telephone number up there in the title of this piece. It is the name of a numerical puzzle guaranteed to spark wonder and amazement in the minds of your students. It is called Kaprekar’s (pronounced kuh-PREE-kur) Constant*. This little, mysterious math activity is one of a family of math procedures called Recurrent Operations.

Recurrent operations are repetitive procedures wherein each intermediate result (i.e. sum, difference, etc.) is used in turn until a specific outcome is ultimately obtained. The procedure to work the Kaprekar sort of “magic” is as follows:

   1.	Select a four-digit number, for example, 5634.

   2.	Form a new number from this number by rearranging the
	digits in decreasing order, e.g. 6543.

   3.	Reverse the number obtained in Step 2, then subtract the
	smaller number from the larger one.

				- 3456

   4.	Take the difference just obtained and repeat the procedure
	in Step 2 and 3; that is, order the digits, reverse, and
	subtract.  This should be continued until the surprise hits

		  8730		  8532		  7641
		- 0378		- 2358		- 1467
		  8352		  6174		  6174

Do you notice that with 6174 the process comes to a “stand-still”? That is, 6174 just yields itself!

But the real surprise is yet to come.

Try the procedure on any other four-digit number. It always ends with 6174. This is a famous number in the field of recreational mathematics; it’s called Kaprekar’s Constant.

Students find this discovery quite fascinating and unexpected, so much so that they don’t even realize that one of your instructional objectives is to provide some practice in subtraction skills. But more importantly you can now use this opportunity as a springboard for further exploration by posing the following questions to them:

   1.	Who can find a number that requires the greatest number
	of subtractions to obtain 6174?

   2.	What happens if the Kaprekar process is applied to three-
	digit or five-digit numbers?  (Dare we ask about still
	larger numbers?)

   3.	What happens if numbers in bases other than ten are used?
	For example, 4-digit numbers in "base 5".

   4.	Why does the game not work on such numbers as 3,333 or 888?

Many things could be said in favor of such activities from a pedagogical viewpoint, but only one will be mentioned here. Due to the strangeness of the whole situation and the interest created thereby, it causes the student to generate his own “problem set” of several exercises, based from only one or two starting numbers. That is to say, by asking him to do “only one or two problems”, he eventually does quite a few individual exercises — and enjoys it all the while.

One final note: if experience in systematic discovery is of more value to your instructional goals than merely subtraction practice, Questions #1 and #2 above make excellent investigations for pocket calculators.

*Discovered by Shri Dattathreya Ramachandra Kaprekar (1905-86?), a mathematician from India. He devised this activity in 1946 or 1949. See his photo on the right.

Additional websites that contain information about this topic:

  1. The Kaprekar Number and more playings with numbers
  2. Mudd Math Fun Facts: Kaprekar’s Constant
  3. Mathematical Black Holes
  4. Scientists of India – D.R. Kaprekar
  5. Sort, Reverse, Subtract
  6. Kaprekar Numbers
  7. More on Kaprekar’s Contants
  8. Mathews: Kaprekar Process (Highly recommended!)

Taken from T. Trotter, “Kaprekar”. Math Lab Matrix, Fall 1976, p. 8.

Distinct Digit Squares


A. When a number is multiplied by itself, the resulting product is called a SQUARE NUMBER, or simply a SQUARE.

12 × 12 = 144 so 144 is a square number.
35 × 35 = 1225 so 1225 is a square number.
133 × 133 = 17,689 so 17689 is a square number.

B. Sometimes a square is made up of digits that are all different, that is, it has “no repeats”. Such a square is called a distinct-digit square (DDS).

Example: 13 × 13 = 169; there are no repeated digits in 169,
so it is a distinct-digit square.

But 21 × 21 = 441; since the 4 is repeated in 441, this is not
a distinct-digit square.


You are to use your calculator to help you make a list of ten (10) distinct-digit squares. But–one more thing–they must all contain either 5 or 6 digits. That is, they should be “5-place” or “6-place” numbers.

Largest Number Squared


If you multiply 142 by itself, what is the product?  _________
If you multiply 781 by itself, what is the product?  _________
Now look at your two answers.
The first one was a 5-place number, and the second one was a
6-place number, right?
(If not, you made a mistake somewhere.  Do the wrong one(s) again.)


You now see than when you multiply a 3-place number by itself, you might get a 5-place or a 6-place product.

Your problem is to use your calculator to find the largest 3-place number that when multiplied by itself gives just a 5-place product.

(Hint: The number is greater than 142.)


Compute these two products:

1022 × 1022 = ________
7803 × 7803 = ________

Do you see that the first product is a 7-place number, and the second one is an 8-place number? (If not, check your work as before.)

This time you are to find the largest 4-place number which when multiplied by itself will still only make a 7-place product.

(HINT: It is greater than 1022.)


Compute these two products:

17 × 17 = _______
83 × 83 = _______

Do you see that the first product is a 3-place number, and the second one is a 4-place number?

This time you are to find the largest 2-place number which when multiplied by itself will still only make a 3-place product.

(HINT: It is greater than 17.)

PROBLEM IVThe Brainbuster

You have done three problems with your calculator that were almost the same. Each time you had to find the largest number which
when multiplied by itself gave a product with an odd number of places,

Now you will be asked to do the whole thing one more time–this is the BRAINBUSTER!

Find the largest five-place number which when multiplied by itself gives only a nine-place product.

But unfortunately, this time your calculator will not be able to help you; a 9-place number is too big for the calculator’s display area.

However, things are not so bad if you will look at the answers you found for the first three problems. There is an important clue there that will tame this tough problem. Do you see it?

The largest 3-place product came from ______;
The largest 5-place product came from ______;
The largest 7-place product came from ______.

Same-Digit Pairs of DDSs


In first section you found several squares that we called DDSs. (Remember: these are squares whose digits are “all different, no repeats”.)

In this section, we will explore something interesting about certain of those DDSs. Look at these squares:

37² = 1369 and 44² = 1936

Both 1369 and 1936 are DDSs, of course. BUT, there is one more thing that is strange: they both contain the same digits, just arranged in a different order.

There are many more cases like this. Before you start the exercise below, make sure you understand this idea by finding the squares for these two numbers: 32 and 49.


In the groups of numbers below, two of them will give DDSs with the same digits, but arranged in a different order. The other numbers also produce DDSs, but do not have the same digits. Find the correct
pair in each group.

  1. 144, 175, 174
  2. 305, 153, 198
  3. 136, 228, 267, 309
  4. 233, 193, 305, 172
  5. 152, 142, 118, 179, 147

Below is given a large group of numbers that will give “same-digit pairs”, like you found above; some will not. Find the numbers that make this type of pair and put them together.

267 281 186 273 224
213 282 286 226 214

Once in a while we can find three or more DDSs that use the same digits. Look at this example:

36² = 1296     54² = 2916     96² = 9216

Do you see that all three squares contain the same digits, only in a different order. Now this is strange indeed! And it does not happen as oiften as was true for the same-digit pairs. But, as we will see, it can happen several times, if we are patient enough to look.

The following eleven numbers will produce DDSs that can be grouped into three same-digit families. Each family will have at least three members in it, maybe more. Can you separate all of them into their proper families?

181 148 154 128
209 203 269 196
191 302 178 .

So far, all of our DDSs have been only 5-place numbers. But the same thing can happen with 6-place DDSs, too. And, would you believe it? There are even more pairs and family-sized groups than you saw before.

Here are several numbers that will produce DDSs pairs or families. Can you separate them as you did before?

324 353 364 375
403 405 445 463
504 509 589 645
661 708 843 905


This piece was written by me and published in The Oregon Mathematics Teacher, Sept. 1978. At that time calculators with a 10-digit display were not the common models available to students at the elementary or middle school levels. So the “Brainbuster” problem above needs to be adjusted to take that into account, or only permit the use of 8-digit models while
doing this activity.

Trotter in Prime Curios


In the latter part of May of this year (2001) we discovered a very interesting website all about prime numbers, titled appropriately The Prime Pages. There is a companion page connected with it, called Prime Curios, a collection of clever and interesting trivia, moderated by G. L. Honaker, Jr. It is to this 2nd site that this WTM page is concerned.

Naturally, we began submitting our own contributions right away. First, we sent the one about 1992 that appears at the beginning. Then others began to follow in rapid succession. The list soon grew rather lengthy, so we decided to place ours in one location. So what you will see and read below is the results of our number play since that time. Nothing is in any particular order, unless otherwise indicated.

Each entry in the list is preceded by a link (the highlighted number) that will take you to the specific page in the website of Prime Curios, so that you may read all the other interesting facts that other people found about that particular number. We think you will be quite surprised and well rewarded.

So go forward now, and most of all, have fun with numbers!

  1. 1992 1992 = 8 x 3 x 83. The only other two years in the period 1000 to 1999 to share the structure of “a x b x ‘ab'” (where ‘ab’ is prime and the concatenation of the factors a and b) are 1316 = 4 x 7 x 47 and 1533 = 7 x 3 x 73.
  2. 1992 If you separate the digits of 1992 like 199 2, you have two primes. Note that 199 is the largest prime less than 2 hundred. Separation in the middle gives us 19 92, and 19 turned upside down along with 92 reversed are prime.
  3. 2310 2310 = 2 x 3 x 5 x 7 x 11 = 112 + 132 + 172 + 192 + 232 + 292. Note the consecutive digits: 0, 1, 2 and 3.
  4. 197 197 is the only three-digit prime Keith number.
  5. 26 The prime factorization of 26 uses the first three counting numbers.
  6. 17 17 is the smallest Trotter prime, i.e., a prime of form 10 x (n2) + 7, where n = 1, 2, 3 …
  7. 1234567 The prime factors of 1234567 (127 x 9721) form a peak-palindromic arrangement of digits (1279721). It is curious that the prime factors of 1279721 are all three emirps (79 x 97 x 167) which upon concatenation form yet another prime (7997167). [Note: see this reference…World of Numbers for more information.]
  8. 36 The smallest square that is the sum of a twin prime pair: 17 and 19.
  9. 8 8 is the smallest cube which is the sum of a twin prime pair {3 + 5}.
  10. 23 2n + 3n is prime for n = 0, 1 and 2.
  11. 23 23 is the smallest prime for which the sum of the squares of its digits is also an odd prime.
  12. 31 The number of letters (in English) required to write the word names of the first six primes is the reverse of the sixth prime (13), namely 31.
  13. 73 73 is the smallest prime whose square (5329) is the concatenation of two multi-digit primes.
  14. 73 389 and 17 are primes, as is their concatenation (38917). Inserting the lowly 0 between them transforms the prime into a power, i.e. 389017, the cube of 73, a prime itself.
  15. 83 The cube of 83 (571787) is the smallest case of the concatenation of a pair of 3-digit primes.
  16. 211151 The smallest Xmas Tree Prime with 3 rows, i.e. 6 digits.
  17. 2111511013 The smallest Xmas Tree Prime of 4 rows, i.e. of 10 digits.
  18. 211151101310867 The smallest Xmas Tree Prime of 5 rows, i.e. of 15 digits.
  19. 3883 This palindrome is transformed into a pal-prime upon the insertion of a single 0 in the middle.
  20. 121661 121661 is the smallest OP-PO Prime.
  21. 72727 72727 is a pal-prime, and separating the digits so — 72 and 727 — we can now say the sum of the digits of the first 72 primes (2, …, 359) is 727, another pal-prime. (P.S. I’m submitting this on 7/27 of this year!)
  22. 277 76729 is the square of the prime 277 and the smallest square with 5 or more digits that is the concatenation of three primes (7, 67, and 29). (Note: the square contains the digits of the root embedded in reverse order.)
  23. 159 159 = 3 x 53, and upon concatinating the prime factors, we have a peak palindrome, 353, which is itself a prime.
  24. 159 Its square (25281) is the concatenation of 2 primes: 2 and 5281.
  25. 77 The square of 77 is 5929, the concatenation of two primes, 59 and 29.
  26. 181 181, a pal-prime, is the sum of the digits of the first 23 primes (2, …, 83).
  27. 217 47 and 89 are primes, as is their concatenation (4789). Inserting the lowly 0 between them transforms the prime into a power, i.e. 47089, the square of 217.
  28. 311 311 is the 11th three-digit prime for which the sum of the squares of its digits is also a prime; and the sum here is 11 as well.
  29. 2357 21 + 33 + 55 + 77 and 22 + 33 + 55 + 77 are twin primes. [Trotter, Kulsha]
  30. 2357 Letting A=1, B=2, …, Z=26, then 2357 is the sum of all the values of the U.S. Presidents’ last names from Washington to Coolidge. [Ref. Wordsworth.]
  31. 2357 2357 is also the sum of consecutive primes in at least two ways: (773 + 787 + 797) and (461 + 463 + 467 + 479 + 487).
  32. 17 Using A = 1, B = 2,…, Z = 26, 17 is the smallest non-negative number whose numerical value of its word form is also a prime (109).
  33. 7 Using A = 1, B = 2, …, Z = 26, the sum of all the letter values in the word names of the numbers from 1 to 7 is a prime (367). If 0 is included, the sum is yet another prime (431).
  34. 64 Using A = 1, B = 2, …, Z = 26, the sum of all the letter values in the word names of the numbers from 1 to 64 is a prime (7369). If 0 (whose alpha-numeric value is 64) is included, the sum is yet another prime (7433).
  35. 5 Only 5 U.S. Presidents have 5 letters in their last names (Both Adams, Grant, Hayes, and Nixon).
  36. 23 23 = 14 + 23 + 32 + 41 + 50.
  37. 13 Using the first three primes we have: 23 + 5 = 13.
  38. 111 111 equals the sum of 2 + 3 + 4 + … + 17 minus the sum of the primes less than 17.
  39. 131 131, a palindromic prime, equals the sum of 2 + 3 + 4 + … + 19 minus the sum of the primes less than 19.
  40. 414 The exponential factored form of 414 (2 x 32 x 23) consists of three 2’s and two 3’s; whereas its expanded form (2 x 3 x 3 x 23) has two 2’s and three 3’s.
  41. 434 434 is also the sum of the cubes of the digits of the emirp 347.
  42. 821 The smallest prime of the first prime quadruple for which the sums of the cubes of the digits of the 4 primes (821, 823, 827, 829) are primes themselves (521, 547, 863, 1249).
  43. 440 The sum of the first 17 primes (2 to 59) and also the number of yards in the quarter-mile race in track-and-field competitions.
  44. 997 The largest 3-digit prime AND the sum of the cubes of its digits is also a prime (1801).
  45. 137 The sum of the squares of the digits of 137 is 59, another prime, and all five odd digits are used (Ref. Father Primes).
  46. 317 The sum of the squares of the digits of the prime 317 is 59, another prime. Note that all odd digits are present.
  47. 165 165 is a multiple of (16 – 5), which is its largest prime factor.
  48. 132 The concatenation of its three distinct prime factors (2, 3, and 11) forms primes in three ways: 2311, 2113, and 1123.
  49. 131143 This prime is composed of three 2-digit primes — 13, 11, and 43.
  50. 123456789 Replacing each of the digits, one-by-one with a 0, yields primes in three cases: 1, 2, and 7 (023456789, 103456789, and 123456089). Note that 127 is a Mersenne prime.
  51. 37 37 + 4n yields primes for n = 1, 2, 3, 4, 5, 6, 7.
  52. 45 If the first five powers of 2 (2, 4, 8, 16 & 32) are each subtracted from 45 all results are primes (43, 41, 37, 29 & 13).
  53. 170 The 170th Trotter number (289007) has an all-emirp prime factorization: 37 x 73 x 107. Note: the 3rd prime is a permutation of the digits of the original number.
  54. 304589267 A prime containing 9 distinct digits, where upon inserting symbols [30/45 + 89/267], we discover the missing digit “1”. The only other prime for which this is possible is 536948207. [Trotter and Knop]
  55. 13831 13831 is the smallest multi-digit palindromic prime such that the sum of it with the next prime (13841) is a palindrome (27672).
  56. 10501 A palindromic prime that is the sum of 3 consecutive primes (3491, 3499, and 3511), while at the same time serving as the middle prime of a set of three consecutive primes whose sum is another palindromic prime (31513).
  57. 97679 96769 is the largest 5-digit palindromic prime that is the sum of 3 consecutive primes (32251 + 32257 + 32261).
  58. 94949 The only 5-digit palindromic prime that is undulating and the sum of 3 consecutive primes (31643 + 31649 + 31657). Note that by adding 3 consecutive primes we only get one other undulating palindrome (16161), which is a non-prime.
  59. 98789 The largest 5-digit palindrome that is the sum of 3 consecutive primes (32917 + 32933 + 32939). Its prime factorization is 223 x 443 and 223 + 443 = 666!
  60. 13124…97909 (24-digits) A prime composed of eight 3-digit palindromes of a “consecutive” style, and one-nineth of (118)8 … 1.
  61. 11111117 11111117 and 71111111 are both primes, thus emirps.
  62. 742950290870000078092059247 (27-digits) The first prime in an arithmetic sequence of 10 palindromic primes. It was found by Dubner and his assistants and has common difference of 10101 x 1011.
  63. 144169 It is also the concatenation of three squares (144, 16, and 9). Note that: sqrt (144) = sqrt (16) x sqrt (9). [Note: this is an extension to another person’s “curio”.]
  64. 8609The largest distinct-digit pime. Pimes (pronounced with a long i) are primes whose digits contain circles, i.e., using only the digits 0, 6, 8, 9. Note: 6089 and 8069 are also distinct-digit pimes.
  65. 174 174 = 72 + 53 (using the first four primes).
  66. 199 199 is also a Permutable prime, meaning that 919 and 991 are primes as well.
  67. 2213 2213 is a “sum of cubes” as follows: 23 + 23 + 133.
  68. 2222 The smallest number divisible by a 1-digit prime, a 2-digit prime, and a 3-digit prime.
  69. 1429 The prime sum of two famous baseball records: 714, number of homeruns hit by Babe Ruth, and 715, number of the homerun hit by Hank Aaron to break the Babe’s record (on 4/8/1974).
  70. 202 A semiprime palindrome equal to (2 + 3 + 5 + 7)2 – (22 + 32 + 52 + 72). It’s the only such case for all primes < 2,000,000,000. [Trotter and De Geest]
  71. 576 A square equal to (2 + 3 + 5 + 7 + 11)2 – (22 + 32 + 52 + 72 + 112). It’s the only such case for all primes < 2,000,000,000. [Trotter and De Geest]
  72. 223 The sums of the nth powers of its digits are prime for all n between 1 and 6 inclusive: sum of digits = 7, sum of squares of digits = 17, sum of cubes of digits = 43, sum of fourth powers = 113, sum of fifth powers = 307 and sum of sixth powers = 857.
  73. 607565706 A palindrome resulting from this “prime based” expression: (257 + … + 607)2 + (2572 + … + 6072). There are 57 consecutive primes inside each parentheses. Note that this palindrome starts with the last prime added: {607}565706. “57” appears also as a substring, 60756{57}06. The number of the beast {6}075{6}570{6} is included. [De Geest and Trotter]
  74. 107 Rudy Giuliani was the 107th mayor of New York City.
  75. 2003 There is only one way to use consecutive integers to produce a sum of 2003.
  76. 20022002 The prime factorization of 20022002 is 2 x 7 x 11 x 13 x 73 x 137, which when grouped thus, 2 x 11, 13 x 73, and 7 x 137, yield 3 palindromic semi-primes: 22, 949, and 959.
  77. 1951 1951 is prime, and appears as the 9th term of the sequence 1+9+5 = 15, 9+5+15 = 29, 5+15+29 = 49, etc.
  78. 98689 The first centered triangular number (i.e. of the form the form of (3n2 – 3n + 2)/2) that is a palindromic prime.

This next group of 7 items is the result of some email correspondence we had with Carlos Rivera (6/27/01). [See Potpourri for that email.] We posed the basic idea, and Carlos provided us with the numbers. (See above on 36 and 8 for the cases of square and cube.)

  1. 253124999 The smaller of the smallest twin prime pair for which the sum is a 4th power (sum = 1504).
  2. 4076863487 The smaller of the smallest twin prime pair for which the sum is a 5th power (sum = 965).
  3. 578415690713087 The smaller of the smallest twin prime pair for which the sum is a 6th power (sum = 3246).
  4. 139967 The smaller of the smallest twin prime pair for which the sum is a 7th power (sum = 67). (Note: this prime ends with a 6 and 7.)
  5. 14097…72287 (26-digits) The smaller of the smallest twin prime pair for which the sum is an 8th power (sum = 15188). [The entire number is 140975 6730907423 9886172287.]
  6. 73099303486215558911 The smaller of the smallest twin prime pair for which the sum is a 9th power (sum = 1749).
  7. 8954942912818222989311 The smaller of the smallest twin prime pair for which the sum is a 10th power (sum = 16810).

The next items are the results of looking at someone else’s “curio”, then expanding a little on it. We suggest that you go to the Prime Curio page to see the full number and the name of the original submitter.

  1. 15555…55551 (33-digits) The digital sum of this prime is 157, another prime (whose digit sum in turn is yet another prime: 13).
  2. 10220…02201 (55-digits) The digit sum of this prime is 110, which is the double of its number of digits.
  3. 14444…44441 (67-digits) The digit sum of this prime is 262, a peak palindrome.
  4. 18181…18181 (77-digits) The digit sum of this 77-digit prime is 343, the cube of 7.
  5. 31313…31313 (83-digits) The digit sum of this prime is the prime 167.
  6. 19999…99991 (87-digits) The digit sum of this prime is a palindrome, 767.
  7. 37777…77773 (87-digits) The digit sum of this prime is 601, another prime.

Finally, we present some items that are harder to categorize. The first one was created by the moderator of Prime Curios, G. L. Honaker, Jr., after we wrote our webpage on Trotter Numbers and Trotter Primes. The second one is a two-person contribution, involving Monte Zerger (whose name and creations can be found in other WTM pages) and ourselves (WTM).

  • 735 There are exactly 735 Trotter primes less than 100,000,000. Note the first three odd primes in 735. [Honaker]
  • 510 The concatenation of 510 with itself (510510) is the product of the first 7 primes and also the product of the 7th through 10th Fibonacci numbers (13, 21, 34, and 55). [Zerger](Continuing the previous curio) The difference between the next prime (19) and the next Fibonacci number (89 – also a prime) is 70, which is the product of the Fibonacci subscripts above. [Trotter]
Note: Some of the above items have been removed from the Prime Curios page, though they at one time were indeed posted. Still it doesn’t alter the basic facts about any given entry. It was merely a decision taken later by the site moderator.

Rep-Digit Numbers

This is a math activity that needs you to use your best investigation skills. So get out your paper and pencil and put on your thinking cap. (And you may wish to consider a calculator, but it won’t be of much use some times.)


Let a rep-digit number be any number that consists of a string of only one kind of digit, but which may be repeated as much as desired. Hence, the name rep-digit is short for repeated-digit.

For example, 111, 44444, 66, 22222222, and 8 are all rep-digit numbers. Their lengths (or if you prefer, orders) are 3, 5, 2, 8, and 1, respectively.

Problem #1

Show that any rep-digit number made up of ones (i.e. 1, 11, 111, etc.) when divided by 7 leaves a zero remainder when the length of the number is a multiple of six; otherwise such divisions will leave any remainder (less than 7) except 3.

Problem #2

What corresponding peculiarities can you discover for the remainders when you divide other rep-digit numbers by 7. That is, sets of numbers of the form dddddd, … , where d can be any whole number from 2 to 9.

Problem #3

Now combine the ideas of #1 and #2 and use other numbers besides 7 as the divisors. What patterns can you find? What overall observations can you make? Pay special attention to divisors that are primes.

I especially recommend the primes 13, 17, 19, and 23. Make a string of 1’s long enough until you have no remainder at all with those numbers as the divisors. If you begin by putting the “unnecessary” 0’s in the quotient position, it will help in discovering some unique facts of the “split-half” variety. (See the number 142857 in Special Numbers.)

		           0 0 8 5     
			13) 1 1 1 1 ...
		    	    1 0 4
				7 1
			      6 5

One last piece of advice: if you’re doing this long division work “by hand”, use a piece of ordinary graph paper, and put one digit in each square as you move through the procedure. It helped me, so I think it would help you. 😉

Digital Diversions

A Digital Diversion

Perform the following steps as indicated to see an interesting result.

  1. Form the smallest possible 4-place number using the four largest digits.
  2. Add that number to itself. The result is sum #1.
  3. Take sum #1 and add it to itself. The result is sum #2.
  4. Finally, take sum #2 and add it to itself. The result is sum #3.
  5. While all 3 sums are interesting & share a common property, sum #3 is probably the most unique of all, especially when compared with the original 4-place number.

What is that strange aspect?

Do you care for another similar trick? Then try this:

  1. Form the largest possible 4-place number using the four largest digits.
  2. Divide that number by 4.
  3. Multiply the quotient just obtained by 5.
  4. Observe the resulting product carefully.

Aren’t numbers marvelous?

Another Digital Diversion

It is a well known, and easily proven fact, that using 4 distinct digits, such as 3, 7, 2, 4, and so on, you can form 24 different 4-place numbers. The same principle would apply using 4 different letters of the alphabet, namely you could form 24 different “words”. The words need not make sense or even be pronounceable. Likewise, .you could arrange 4 persons in four chairs in a row in 24 different ways. Such an arrangement is called a permutation of the 4 items under consideration.

Now to continue in the line of reasoning of our previous digital activity…

Take the digits 2, 3, 7, & 9, and form all 24 possible 4-place numbers. It is our goal now to find which of those numbers will yield for us the remaining five digits (1, 4, 5, 6, & 8) when they are doubled.

Extra 1: While you still have your list of 24 candidates handy, try this with those that did not work out: multiply by 5 in order to get products formed by the same five digits (1, 4, 5, 6, & 8).

Extra 2: Now replace the 3 above with a 6 and repeat the above doubling & quintupling process on those 24 permutations. That is, use 2, 6, 7, & 9 to yield results containing only the digits of 1, 3, 4, 5, & 8.

A Digital Diversion a la Kaprekar

A little known piece of number trivia concerns a mathematician from India, named D. R. Kaprekar. He discovered that if you use the digits 1, 4, 6, & 7 to make the largest and smallest 4-place numbers, that the difference between them is a number that is composed of those same four digits. This is easily shown as follows:

7641 – 1467 = 6174

The number, 6174, is therefore called Kaprekar’s Constant. Much has been written about this idea in various websites*, math books & periodicals.

However, your task here is (1) to list all 24 permutations of those digits as 4-place numbers; then (2) use some other number(s) to multiply them by in order to obtain the remaining five digits (2, 3, 5, 8, & 9). [Hint: the factors you need are less than 10.]

Extra: three of the permutations will yield good results when multiplied by 12, 16, & 53.

Note: use of a spreadsheet is recommended for this problem.

[*For my website, go to Kaprekar. There are other links there as well.]

DD’s Two-by-Two

By now, you should be good at listing the 24 permutations of 4 distinct digits to form 4-place numbers. Also you should be pretty good at finding which permutation(s) yield the remaining unused digits to form the final result.

So without further ado, we will give you nine sets of 4 digits to work with. Find which permutations produce the other five digits in their products when multiplied by 8.

(If you think about it for a moment, that’s actually what you were doing in the problem task at the beginning of this collection of activities. That is, doubling something three times in sequence is equivalent to multiplying by 8: N * 2 * 2 * 2 = N * 8.]

Here are the sets. Each set will produce 2 good products. Sorta like Noah putting the animals in the ark 2-by-2, isn’t it?

            {1, 2, 3, 7}    {1, 4, 5, 9}    {1, 4, 6, 9}

            {1, 5, 6, 9}    {2, 3, 5, 9}    {3, 4, 8, 9}

            {3, 5, 8, 9}    {4, 5, 8, 9}    {4, 6, 7, 9}

Kaprekar Revisited

Above in another activity you were introduced to the set we call Kaprekar’s Digits. This time we will insert the digit 0 and call the set the augmented Kaprekar set.: {0, 1, 4, 6, 7}. You will be multiplying by numbers formed from that set: e.g. 10746, 41067, etc. As before, the set for the products will be {2, 3, 5, 8, 9}.

Now the number of permutations for 5 things is 120, which is a bit boring to list out. Therefore, we will change our approach this time. Your task will be in the form of a traditional matching quiz. This means, we will list the permutations that work in the left column and the other factor in the right column. Once a match has been made, those numbers can be set aside and considered as completed.

There is another difference this time. The second factor will not be a whole number. Instead it will be non-integral, greater than 1. An improper fraction if you prefer. This proved to be necessary due to the nature of the digits used.

So let’s go. Match them up!

permutation factor
67014 8/3
70146 6/5
67140 9/8
71460 16/7
74610 17/15
41067 9/4
14607 4/3
61470 17/4
17460 5/4
47016 4/3
41706 7/3
14076 19/4
46710 8/5
17460 5/4

Pandigitals and Fibonacci

Surely you are familiar with a famous category of numbers in the mathematical world, called the Fibonacci Numbers. If not, we recommend that you do a search of the web and you will be amazed with this fascinating topic. Suffice it to say here that they are those numbers that appear in this sequence

1, 1, 2, 3, 5, 8,13, 21, 34, 55, etc.

where each number after the first two is found by adding the 2 preceding numbers. Study that example to convince yourself that it’s true.

Now we plan to connect the idea of the Fibonacci addition procedure and pandigitals in a clever and interesting way. Here’s how:

We will start with a certain number – it may be special in its own way, like being a palindrome, a prime, etc. – then we will try to get a pandigital result sooner or later by applying the special method of addition. Watch this:

Let’s start with the palindrome 9530359. Then add 9530360, the next counting number after it. The result is 19060719. Now add 9530360 and 19060719, to obtain 28591079. Continuing like this, our next two sums are 47651798 and 76242877. When those two are finally added, — voila! – we have our pandigital of 123894675. This only took us 5 steps of hard adding. Not bad, eh?

To avoid unnecessary re-copying of these big numbers, we could condense our presentation like so:


+ 9530360











Looks rather easy now, doesn’t it? Well, we’re just getting started. Here are some more starting numbers for you to play with. Some take more steps to arrive at the “pandigital 9″, some require fewer. So be careful. One little slip and you’ll be going off in the wrong direction.

  1. 4187814
  2. 4870784
  3. 6097906
  4. 630036
  5. 6834386
  6. 4004
  7. 82466428
  8. 1993  (a prime year)
  9. 102013  (another prime)
  10. 30013  (a "lucky" number)


The activities above are based on a common theme – using the digits from 1 to 9, once and only once, to produce a special effect. It is a popular theme that has many variations, sometimes including the 0 as well. Some of the other variations are in other pages of WTM, while still others in other books and websites. For the individual who may be interested, we will give the web links here.

From the WTM:

  1. Double/Triple
  2. Nine-Digits-&-Equal-Products
  3. Ticket Number Trivia
  4. More Strange Multiplication
  5. Five Distinct Digits
  6. S.T.T.H.W.N. (Part II.2; Part IV.1)
  7. Special Numbers II (99066)
  8. FUN for Figurenuts

From the World!OfNumbers:

  1. http://www.worldofnumbers.com/em107.htm
  2. http://www.worldofnumbers.com/em114.htm
  3. http://www.worldofnumbers.com/em137.htm
  4. http://www.worldofnumbers.com/ninedigits.htm
  5. http://www.worldofnumbers.com/em85.htmM