Dottable FractionsMike Keith

July 1998

IntroductionA

dottable fractionis 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=4·1621879 2·187·9Note 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 fractionsdiscussed in [1], [2], and [3] are a special case of this general formulation.We refer to a dottable fraction with

Nnumerator digits andDdenominator digits as being oftype N/D. If the numerator is separated by the dots into chunks ofn_{1}n_{2}...n_{k}digits and the denominator intod_{1},d_{2}...d_{m}, we say that the fraction is ofsubtype n_{1}n_{2}...n_{k}/d_{1}d_{2}...d_{m}. 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 haveN-1 things which can be in either of two states, there are 2^{N-1}possibilities; but since we require at least one dot to be present this eliminates one arrangement, so the correct number is 2^{N-1}-1. Similarly, there are 2^{D-1}-1 cases for the denominator. We can use all possible arrangements of numerator and denominator dots, therefore there ares(

N,D) = (2^{N-1}-1) x (2^{D-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 FractionsFrom a dottable fraction of type

N/Dand subtypen_{1}n_{2}...n_{k}/d_{1}d_{2}...d_{m}one can construct a larger dottable fraction of typeN+r/D+s, and subtypen_{1}n_{2}...n_{k}+r/d_{1}d_{2}...d_{m}+sby placingrzeros on the end of the numerator andszeros on the end of the denominator (for anyr,s> 0). Therefore, in this paper we only considerprimitivefractions, 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/dis dottable, so is its reciprocald/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 ifN=Dthen 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,cand the denominator digits bed,e,f,g. Then for the fraction to be dottable we must have(100

a+ 10b+c) / (1000d+ 100e+ 10f+g) =a·(10b+c) / (d·e· (10f+g))which is equivalent to the somewhat unwieldy Diophantine equation

1000

adef+ 100adeg+ 100bdef+ 10bdeg+ 10cdef+cdeg=

10000abd+ 1000acd+ 1000abe+ 100aec+ 100abf+ 10acfwhich is to be solved under the constraints (for primitive solutions) that

a, c, d, andgare integers between 1 and 9 andb,e, andfare 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 forN,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/Dfraction is dottable using this subtypebut rather as

for each type-

N/Dfraction

find all ways (subtypes) in which this fraction is dottableThe 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 many12 / 12 812 / 21 1212 / 111 3721 / 12 2321 / 111 17111 / 12 16111 / 21 1111 / 111 59Total173Two 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 Subtypes2 2 none2 3 none2 4 none2 5 11/413 3 21/213 4 21/314 4 31/31 121/31 1111/314 5 31/41 31/32 11111/415 5 32/41 32/32 41/41 41/32 41/311 41/11111 311/41 311/32 1121/41 2111/41 11111/41Table 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 subtypestends 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:(10

^{r}a+b) / (10^{s}c+d) =ab/cdwhere

banddhaverandsdigits, respectively. Roughly speaking, ifrandsare "small", the effect ona/bof multiplication byb/don the right-hand side of this equation will always be too large, compared to the effect of addingaandbto the numerator and denominator of the left-hand side, for equality to be possible. A theorem in [3] states that ifN=Dandr=s, this happens if and only ifr£ [(N-1)/2]; similar theorems can be derived for the case whereN<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 5D: 27353 1564127 1219 33645323 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 / 3718512069 / 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 remarkable666=6·6·664676 6·46·76in which 666 appears in the numerator

andin 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.