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Desiring to use such fascinating problems in my school teaching at the junior high level, I wondered, "Are there more such problems, or is this just a freak case?"
The Solution
To my great surprise, once the simple secret of their construcion is understood, I found it very easy to find quite a few such examples. But, often one has to be rather patient for the required factors to reveal themselves. To facilitate the following discussion, I will demonstrate
the construction method with a simple example. (This will also serve to
motivate the rationale behind the recurrent operation to be described shortly.)
Say that we desire to have a problem where "4" is our single-digit multiplier, and "7" is to be the shifted unit's digit. We begin by setting those digits in their normal places (see Figure 1a).
2 2 3 2
7 _ 7 8 7 _ 8 7
× 4 × 4 × 4 × 4
8 8 4 8
(a) (b) (c) (d)
Figure 1
The indicated multiplication is carried out as shown in Figure 1b. Then as the "8" is the unit's digit of the product, it follows that it must be the ten's digit of the factor as well (see 1c). Figure 1d shows how the process is to be repeated.
But when does it stop? It's quite simple: when you obtain a "7" to place in the product and there is no "carry" involved. In the example before us, we are fortunate to arrive at another "7" in only six steps:
3 3 1 3 2 1 7 9 4 8 7 × 4 7 1 7 9 4 8In general then the process continues until such time that the shifted digit itself is the outcome of one of the steps.
By using this procedure for other original digit choices, one can produce multiplication problems that possess the back-to-front shift property for the multi-digit factor.
| Table 1. Back-to-Front Multiplication Integers | |
| Single-digit factor | The Special Integers |
| 2 | 105,263,157,894,736,842 |
| 3 | 1,034,482,758,620,689,655,172,413,793 |
| 4 | 102,564 128,205 153,846 179,487 230,769 |
| 5 | 102,040,816,326,530,612,244,897,959,183,673,469,387,755 142,857 |
| 6 | 1,016,949,152,542,372,881,355,932,203,389,830,508,474,- 576,271,186,440,677,966 |
| 7 | 1,014,492,753,623,188,405,797 1,159,420,289,855,072,463,768 1,304,347,826,086,956,521,739 |
| 8 | 0,253,164,556,962 0,379,746,835,443 0,886,075,949,367 1,012,658,227,848 1,139,240,506,329 |
| 9 | 03,370,786,516,853,932,584,269,662,921,348,314,606,741,573 10,112,359,550,561,797,752,808,988,764,044,943,820,224,719 |
Table 1 presents a summary of what I choose to call the "primary" solutions. These solutions were developed in the following systematic manner:
1) First I selected various digits (destined to be the shifted
unit's digits) that were equal to or greater than certain
single-digit multipliers.
2) While computing all such cases, I observed that frequently
a result contained the same sequence of digits; it just
began at a different position. Here is a simple example
to illustrate:
1 2 8 2 0 5 2 0 5 1 2 8
× 4 × 4
5 1 2 8 2 0 8 2 0 5 1 2
So, the "shifted-8" case was considered redundant for my
purposes, and was not included in the table. This should
make it clear why there are fewer integers listed than are
possible, i.e. the others are merely "embedded" in those given.
3) There were only a few cases where this embedded aspect did
not account for instances in which the shifted-digit was less
than the multiplier. This happened only when "8" and "9"
were the multipliers, occurring three times and once,
respectively. It is in such situations for all multipliers
that a "leading zero" must be appended to the front of the
large factor. The reason for this becomes obvious when it
is pointed out that no positive integer times another can be
less than the former one (whether or not there is a "carry").
In view of the way it makes things more uniform and regular,
the use of the leading zero is not too great a liberty to
take with standard notation., one thing is clear at this
point: it preserves the equal length property of the integers,
a fact that will become more apparent later.
Since further structure and pattern considerations for the integers in Table 1 will actually be covered in the discussion of the recurrent operation, we should now turn our attention to that topic.
3 3 1 3 2 3 3 1 3 2
1 7 9 4 8 7 1 7 9 4 8 7 /×4/
× 4 × 4 7
7 1 7 9 4 8 28
34
19
(a) (b) 37
31
(7)
(c)
Figure 2
The recurrent operation algorithm then emerges in the 2c part, by observing these steps:
This can be summarized by the recursive formula:
Let k and a1 be the constant and first-term digits, respectively. Then,
a2 = ka1 , and an = ku + t , where an-1 = 10t + u.Table 2 was prepared by using this compact vertical format. Reading the unit's digits from the bottom up produces the special integers given in Table 1. Table 3 (further below)is a summary of the patterns that can be found in Table 2.
| Table 3. | ||||
| Factor | No. of primary cycles | Cycle Period | Max. Integer | Omissions from primary cycles |
| 2 | 1 | 18 | 18 | none |
| 3 | 1 | 28 | 28 | none |
| 4 | 5 | 6 | 37 | 13, 14, 17, 23, 25, 29, 35 |
| 5 | 2 | 42 & 6 | 48 | none |
| 6 | 1 | 58 | 58 | none |
| 7 | 3 | 22 | 68 | 23 & 26 |
| 8 | 5 | 13 | 77 | 12, 14, 15, 17, 27, 33, 41, 57, 58, 61, 69, 71 |
| 9 | 2 | 44 | 88 | none |
This summary table reveals several intriguing properties for the cycles in Table 2. The first thing to note is that the multipliers 2, 3, and 6 each produce but a single primary cycle, whereas the remaining factors yield 2 or more cycles apiece. Turning to the multi-cycle cases, we see that all except the factor "5" had cycles of equal periods. The term "max. integer" simply refers to the largest integer that appeared in the normal production of the primary cycle for the factor. For the factors 2, 3, 6, and 9, it is relevant to note that the product of the cycle period and the number of cycles is equal to the "max. integer". This shows that there are no "omissions" in the range of 1 to that largest integer. The factor of 5 also qualifies here by merely adding the two values for its cycle periods.
However, the factors 4, 7 and 8 present the most interesting facets so far. Number 7 has the fewest omissions, so let's examine them first. Applying the R.O. algorithm. It so happens that each integer (23 and 46) yields itself immediately. In line with current terminology in such cases, I call such integers "self-generators".
Testing the values in the omissions list for "8" shows that none of them are self-generators. However, all but one of them (i.e. 69) generate another entry in the list. Yet 69 itself generates an interesting value nonetheless: "78", which is one greater than the max. integer. So, if we temporarily include 78, we can construct a "secondary" cycle, namely,
This augmented cycle now has 13 terms, just like the primary ones!
Finally, we look at the case for "4". That set has two self-generators (13 and 26). (At this point we can note a common property shared by our two pairs of self-generators: namely, the smaller one is half the larger.) As we did for the 8-case above, another secondary cycle can be formed by using the remaining values. This time we must also temporarily include the integer that is one greater than the max. integer. This last cycle is
And the equal-period property is preserved!
Now, upon glancing back to Table 3, we can appreciate the full significance of the max. integer column by noting that only the "4" and "8" cases had a largest entry ending in a "7"; and only those two yielded secondary cycles that necessitated the restoration of the "missing link", so to speak. With the inclusion of those two missing links, everything "ends with an 8".
That strange finding naturally raises the question: "What about all those values that immediately follow those 8-enders, namely, 19, 29, 39, ... , 89?" Well, oddly enough they all turn out to be self-generators for their respective single-digit factors! And this fact extends beyond the table, and is easily proved as follows:
(The final expression is the same as what is inside the brackets.)
Also, it is very easy to see that all the integers (no matter how large), under this R.O. procedure reduce to only the cycles or self-generators given in this paper. This is a factor that parallels the well-known R.O. procedure concerning the sums of the squares of the digits of all integers (see the Steinhaus or Trigg references). [Also click on Happy & Dizzy Numbers for more information in this website.]
Finally, a challenge to the reader. What patterns, and periods, and other phenomena occur when integers in other bases are put through this new recurrent operation? Try them, and see for yourself!
References
[Published in The Oregon Mathematics Teacher, Feb. 1978, pp. 22-27.]
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