Dottable Fractions

Mike Keith
July 1998


 

Introduction

A dottable fraction is an ordinary decimal fraction, like 416/21879, in which one or more dots (representing multiplication) can be inserted in both the numerator and denominator to make an expression equal in value to the original fraction. For this fraction, such an expression is:

 416  =    4·16   
21879    2·187·9

Note that the locations of the dots need not be the same in the numerator and denominator, and even the number of dots can be different. The multiplicative fractured fractions discussed in [1], [2], and [3] are a special case of this general formulation.

We refer to a dottable fraction with N numerator digits and D denominator digits as being of type N/D. If the numerator is separated by the dots into chunks of n1 n2 ... nk digits and the denominator into d1, d2 ... dm, we say that the fraction is of subtype n1n2...nk / d1d2...dm. The example above is of type 3/5 and subtype 12/131.

For a given type, how many subtypes are there? In the numerator, there are N-1 locations where dots can either be placed or not placed (between each pair of adjacent digits). Since we have N-1 things which can be in either of two states, there are 2N-1 possibilities; but since we require at least one dot to be present this eliminates one arrangement, so the correct number is 2N-1-1. Similarly, there are 2D-1-1 cases for the denominator. We can use all possible arrangements of numerator and denominator dots, therefore there are

s(N,D) = (2N-1-1) x (2D-1-1)

possible subtypes of the type N/D. This fact is of interest because we will see later than certain subtypes of a given type do not yield any dottable fractions, so the actual number of possible subtypes may be less than this value s(N,D).

 

Primitive Dottable Fractions

From a dottable fraction of type N/D and subtype n1n2...nk / d1d2...dm one can construct a larger dottable fraction of type N+r/D+s, and subtype n1n2...nk+r / d1d2...dm+s by placing r zeros on the end of the numerator and s zeros on the end of the denominator (for any r,s > 0). Therefore, in this paper we only consider primitive fractions, in which the units digit of the numerator and denominator are both non-zero. We also do not permit fractions with leading zeros (like 0123/56), so in a primitive solution both the numerator and denominator start and end with a non-zero digit.

Also, if a fraction n/d is dottable, so is its reciprocal d/n (by simply using the "reciprocal" of the subtype). So we also define primitive solutions to always have a value less than 1; that is, N £ D, and if N=D then the fraction must be less than 1. The basic question we consider is: what are all the primitive dottable fractions for a given type or subtype?

First, observe that finding all dottable fractions with a given type and subtype amounts to solving a non-linear Diophantine equation. Take subtype 12/112 as an example. Let the numerator digits be a,b,c and the denominator digits be d,e,f,g. Then for the fraction to be dottable we must have

(100a + 10b + c) / (1000d + 100e + 10f + g) = a·(10b + c) / (d · e · (10f + g))

which is equivalent to the somewhat unwieldy Diophantine equation

1000adef + 100adeg + 100bdef + 10bdeg + 10cdef + cdeg =
10000abd + 1000acd + 1000abe + 100aec + 100abf + 10acf

which is to be solved under the constraints (for primitive solutions) that a, c, d, and g are integers between 1 and 9 and b, e, and f are between 0 and 9.

We used a computer program to search for all primitive dottable fractions, of all types and subtypes with N,D £ 5, by solving the Diophantine equations via exhaustive search (a relatively inefficient method, but one that’s sufficiently fast for N,D £ 5). Two programming subtleties worth noting are:

- To work for N,D £ 5 we have to deal with integers up to 10 decimal digits in size. Since 32-bit integers can only represent up to all 9-digit integers, double-precision floating-point math was used. Even on a personal computer this executes fast enough to be reasonable.

- Since there are many subtypes to be checked for each type, we organized the computations not as

for each subtype
  determine if each type-N/D fraction is dottable using this subtype

but rather as

for each type-N/D fraction
  find all ways (subtypes) in which this fraction is dottable

The second structure allows many computations to be reused, thus speeding up the procedure.

For example, here are the results for type 3/3. There are exactly 156 primitive dottable fractions, but some of them are dottable in more than one way (i.e., using more than one subtype), so if we count the "dottable expressions" there are actually 173, which break down as follows:

Subtype	    How many
12 / 12		8
12 / 21		12
12 / 111	37
21 / 12		23
21 / 111	17
111 / 12	16
111 / 21	1
111 / 111	59
Total		173

Two noteworthy fractions of type 3/3 are 388/485 (= 3× 8× 8/48× 5), the unique one of subtype 111/21, and 148/666 (= 1× 48/6× 6× 6), which is the smallest beastly dottable fraction (one whose numerator or denominator is 666), and the only beastly one of type 3/3.

The other surprise contained in the table above is that there are no primitive dottable fractions with subtype 21/21. This is one instance of a general phenomenon: not all subtypes of a given type can produce primitive dottable fractions. The following table shows the missing subtypes for all types with N,D £ 5:

N       D      Subtypes
2	2	none
2	3	none
2	4	none
2	5	11/41
3	3	21/21
3	4	21/31
4	4	31/31	121/31	1111/31
4	5	31/41	31/32	11111/41
5	5	32/41	32/32	41/41	41/32	41/311	41/11111  311/41
                311/32	1121/41	2111/41	11111/41
Table 1. The "forbidden subtypes": ones for which there are no dottable fractions.

Note that the right-most digit in the numerator and denominator of these forbidden subtypes tends to be small. It is easy to see why this is the case, at least for the two-digit subtypes. In these cases, to have a dottable fraction the following equation must hold:

(10r a + b) / (10s c + d) = ab / cd

where b and d have r and s digits, respectively. Roughly speaking, if r and s are "small", the effect on a/b of multiplication by b/d on the right-hand side of this equation will always be too large, compared to the effect of adding a and b to the numerator and denominator of the left-hand side, for equality to be possible. A theorem in [3] states that if N=D and r=s, this happens if and only if r £ [(N-1)/2]; similar theorems can be derived for the case where N<D, although derivation of a comprehensive theorem to predict all forbidden subtypes seems difficult.

If a subtype is forbidden, this does not necessarily mean that there are no dottable fractions for that subtype. For example, for subtype 311/32 shown in the table above we can produce (non-primitive) dottable fractions by adjoining a zero to the denominator of all the subtype-311/31 fractions (which are the reciprocals of the subtype-31/311 primitive fractions). This is, however, the only such instance in the above table for which we can do this, because for all the others the smaller subtype we might use to construct them is also forbidden.

Here is a summary table of the total number of primitive dottable fractions for each type:

N=   2    3    4    5	
D:
2    7  
3   53  156 	
4  127 1219  3364 	
5  323 2856 22754 58472

Table 2. The number of primitive dottable fractions for each N,D < 5.

Although these numbers are fairly large, dottable fractions appear to be quite rare. For example, the probability that a random 5-digit-over-5-digit fraction is dottable is around 0.00002.

From the type-4/5 and type-5/5 sets we can determine the smallest (in terms of fraction "width") pandigital dottable fractions. There are 24 non-primitive ones that result from adding a zero to the numerator of a type-4/5 one, and 25 primitive ones of type 5/5. These are shown below. Determination of where to place the dots is left as an exercise for the reader!

12960 / 37584
12980 / 74635
13680 / 29754
13950 / 26784
13950 / 46872
17460 / 39285
18270 / 63945
18630 / 27945
28350 / 19764
32160 / 97485
32970 / 16485
34560 / 91728
36450 / 12798
46350 / 12978
46920 / 31875
54270 / 18693
61830 / 92745
68170 / 94235
74180 / 25963
78360 / 21549
79380 / 64512
86310 / 92475
91560 / 73248
92460 / 37185

12069 / 37548
12096 / 73584
13548 / 27096
13608 / 45927
16032 / 48597
17068 / 23594
18074 / 63259
18306 / 27459
18537 / 46092
19602 / 45738
23046 / 79158
24759 / 61308
27018 / 94563
27054 / 93186
30168 / 52794
30186 / 45279
30618 / 45927
30792 / 61584
32064 / 79158
35784 / 69012
40365 / 91287
48516 / 97032
74108 / 92635
79065 / 81324

Table 3: The 49 width-5 pandigital dottable fractions.

In addition to 148/666, mentioned earlier, there are six more beastly dottable fractions with N,D £ 5. The three of type 3/4 are 666/1998, 666/4995 (both subtype 111/1111) and 666/3478 (subtype 111/121). The three of type 3/5 are 666/27972 (subtype 111/1112), 666/38665 (subtype 111/221), and the remarkable

 666  =   6·6·6 
64676    6·46·76

in which 666 appears in the numerator and in every other digit of the denominator!

 

We close with some problems for further research:

- Develop a theorem which predicts as many of the forbidden subtypes as possible.

- Is there a more efficient way (than exhaustive search) of enumerating the dottable fractions?

- Is there a formula for the numbers in Table 2? If not, how about an asymptotic formula?

 

References

[1] S. Kahan, Fractured Fractions, J. Rec. Math., 7(4), pp. 286-292, 1974.

[2] S. Kahan, More Fractured Fractions, J. Rec. Math., 9(2), pp. 101-103, 1977.

[3] M. Keith, Generalized Fractured Fractions, J. Rec. Math., 12(4), pp. 273-276, 1979-80.