There was a post that I made to my Pedagogical Posturing blog in August of 2013 before I created this mathematical blog in the second half of 2015. The title was "Forming Equations from Integer Sequences" and that title was perhaps a little misleading. At the time, I wasn't aware of the OEIS or Online Encyclopedia of Integer Sequences. What I meant was the sequence of digits that define a number. For example, today my diurnal age is 27374 and so the sequence of digits is 2, 7, 3, 7, 4. Concatenation of the digits is not allowed. Thus we cannot have 27, 3, 7, 4 for example.
In that long ago post, I wrote:
Recently I've been using Twitter to create a daily tweet that records my "day count" (number of days I've been alive) plus its factors (if not prime) and some interesting facts about the number itself or one of its factors. Sometimes there's little to say about the number and in such cases I've found that I can usually form an equation by inserting mathematical operators between one or more of the digits.
For example, yesterday the count was 23518 and 23 - 5 = 18. Today the count is 23519 and 2 + 3 + 5 - 1 = 9. I was wondering if it's always possible to create an equation from five digits using the standard mathematical operators (addition, subtraction, multiplication, division and exponentiation in combination with brackets). Obviously with just two digits, it's only possible when the digits are repeated e.g. 99 becomes 9=9. With three digits, it's sometimes possible e.g. 819 becomes 8 + 1 = 9 but generally it isn't e.g. 219. With four digits, it's more likely e.g. 2119 becomes -2 + 11 = 9 but I'm doubtful whether this is always so. There must come a point however, where the number of digits is sufficient to ensure that it's always so. Maybe five digits is that point.
From now on, I'll try each day to form an equation to test out this theory. For example, tomorrow the count is 23520 which becomes 2 + 3 - 5 = 2 x 0 and it works for tomorrow but beyond that let's see.
Needless to say I didn't "try each day to form an equation to test out this theory" but it might be time to give it a go. The 27374 of my diurnal age today is an easy one:$$2 \times 7 -3=7+4$$However, yesterday's number, 27373, doesn't prove so easy. It seems that having two 3's and two 7's in the number makes things difficult. The conditions that I imposed in the original blog post were the use of only the standard mathematical operators of addition, subtraction, multiplication, division and exponentiation in combination with brackets. These operators needed to be applied to the digits in the order in which they occurred.
One modification that I will make here is to allow \(x \, | \,y\) meaning \(x\) is divided into \(y\) as opposed to \(x/y\) meaning \(x\) is divided by \(y\). This seems quite reasonable as it still only involves the operation of division but allows more flexibility. Its application doesn't seem to help in the case of 27373. If we allow the operator \(x\) // \(y\) meaning return the whole number part of the dividend, then an equation is possible:$$2|(7-3)=7//3$$If we allow // then we could allow \(x\) % \( y\) meaning return the remainder as a whole number when \(x\) is divided by \(y\). Thus 7 % 3 = 1. This might be termed modulo division.
I can't see any way to create an equation from 27373 without extending the original conditions. Even concatenation of the digits doesn't seem to help. A common symbol for concatenation is || and thus 2 || 7 = 27. However, I've stipulated that the digits are to be treated as separate so I'll adhere to that condition. The best approach is to stick to the original conditions and if a solution is not possible, then // and % can be resorted to.
In the case of 27374, there is more than one way to create an equation. Here is another way:$$ 2 \times 7 -(3+7)=4 $$So what I'll try to do is to reassert my original goal of trying each day to form an equation and see what patterns emerge.
This activity of forming a "digit equation" is not all that different from one of Quanta's mathematical games called "Hyperjumps". See Figure 1.
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Figure 1 |
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