Introduction

Henry E. Dudeney [1] in his book "Amusements in Mathematics" mentions "The Cab Numbers" in the title of Problem No. 85.

The problem proposed is :

"What two numbers, containing together all the nine digits, will, when multiplied together, produce another number (the highest possible) containing also all the nine digits?. The nought is not allowed anywhere."

The solution given is :

8745231 * 96 = 839542176

The solutions for three digit (two solutions), four digit (six solutions) and five digit (22 solutions) are also given in Dudeney's book and are as follows:

3 * 51 = 153

6 * 21 = 126

8 * 473 = 3784

9 * 351 = 3159

15 * 93 = 1395

21 * 87 = 1827

27 * 81 = 2187

35 * 41 = 1435

2 * 8741 = 17482 ,            2 * 8714 = 17428

3 * 7251 = 21753 ,            3 * 4281 = 12843

3 * 7125 = 21375 ,            3 * 4128 = 12384

6 * 2541 = 15246 ,            8 * 6521 = 52168

8 * 4973 = 39784 ,            9 * 7461 = 67149

51 * 246 = 12546 ,            42 * 678 = 28476

72 * 936 = 67932 ,            14 * 926 = 12964

24 * 651 = 15624 ,            65 * 281 = 18265

65 * 983 = 63895 ,            75 * 231 = 17325

86 * 251 = 21586 ,            57 * 834 = 47538

87 * 435 = 37845 ,            78 * 624 = 48672

Lets Define Cab Numbers

"Cab numbers can be defined as the numbers (consisting of distinct digits excluding 0 ) which can be represented as the product of two numbers which together contain the same digits as the original number"

Computation of Cab Numbers

All solutions for 6,7,8 and 9 digit Cab numbers have been computed by writing a computer program in Fortran. Table 1 gives No. of solutions, Smallest and Largest solution for n-digit Cab numbers for n=3,4,5,6,7,8 and 9.

Table 1

While writing the computer program, following important properties of cab numbers have been used for computation of Cab Numbers:

"The digital root (i.e. sum of digits of a number until single digit is obtained) of the sum of the digital roots of two factors shall be equal to the digital root of the product of two factors".

For example:

6 * 2541 = 15246

Digital root(DR) of 6 is 6.

Digital root(DR) of 2541 is 3 as 2+5+4+1 = 12 and 1+2 = 3

Digital root(DR) of 15246 is 9 as 1+5+2+4+6 = 18 and 1+8 = 9

Digital root(DR) of sum of digital root of two factors = 6 + 3 = 9

Digital root of product of digital root of two factors is also 9 as 6 * 3 = 18 and 1+8 = 9

So in general if X * Y = Z then

Beacuse digits in Z are same as in X and Y both.

To satisfy above condition, there can only be following four possibilities:

So the digital root of two factors can be 2 and 2, 5 and 8, 3 and 6,or 9 and 9 for a Cab number.

Another important property which have been used for computation of Cab Numbers is:

The minimum and maximum sum of digits of a Cab number of n-digits is given by:

Minimum sum of digits = 1+2+3...+n = n*(n+1)/2
Maximum sum of digits = 9+8+7...+(10-n) = 45-[(9-n)*(10-n)/2]

For example for a 6-digit Cab number:

Minimum sum of digits = 1+2+3+4+5+6 = 6*(6+1)/2 = 21
Maximum sum of digits = 9+8+7+6+5+4 = 45-[(9-6)*(10-6)/2] = 39

Similarly between these limits of minimum and maximum sum of digits, the digital root of these sums must be either 4 or 9 as explained above.

For example for a 6-digit Cab number minimum and maximum sum of digits is 21 and 39. To satisfy digital root criteria, the only possible sum of digits of 6-digit Cab numbers can only be 21, 27, 31 and 36.

Table 2 gives minimum , maximum and possible sums for n-digit Cab numbers for n=3,4,5,6,7,8 and 9.

Table 2

All 9-digit Cab Numbers.

There are solutions using each of the ten distinct digits. Many ten digit solutions can be derived from 9 digit solutions by merely appending a zero to one of the factors. But again using computer program it was possible to find all ten digit solutions including zero in an internal position.

There are 4123 solutions obtained with zero in an internal position, out of total 12449 solutions obtained with ten distinct digits(with two factors). Smallest and largest solution obtained are

258 *3967041 = 1023496578

9654 * 871203 = 8410593762

Further investigations

It is interesting to consider more than two numbers(factors) which when multiplied together produce another number containing the same digits as the original numbers. For example:

54 * 38 * 9617 = 123597684

8 * 92 * 531 * 746 = 291548736

6 * 8 * 9 * 71 * 4523 = 138729456

There are total 2900 solutions [3] for 9 digits (i.e. zeroless pandigital) including 1625 solutions for two factors against 2624 total solutions reported earlier [2]. Also there are total 24128 solutions [3] for 10 digit pandigital including 12449 solutions for two factors.

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References:

[1] Dudeney, H. E. Amusements in Mathematics. New York: Dover, 1970.

[2] Madachy, Joseph S. Madachy's Mathematical Recreations. New York: Dover, 1979.