Thursday, November 13, 2014

Detail #120: A Quirk in Numbers

Imagine a numeral system much like the decimal system - although any number really could serve, I will use decimals as the basis here for simplicity. We further keep a detail that is quite common: unique numbers for a few numbers above ten (or whatever base we're using): eleven, twelve, and we give ten, eleven, twelve the symbols Ð, Þ, ß (I only picked those symbols because they were conveniently available right now). These are parsed as their value * 10^(n-1), where n = the index of the position from the left.

Now, partially, English and many other languages do admit using these numbers a bit beyond their usual range: twelve hundred is an example of exactly that. But let's further permit using eleven and twelve beyond that:

1ß1 : 221
ßßß: 1332
ß12: 1212
3Ð2: 402
Ð0: 100
1Ð0: 200
5Þ: 61
Þ5: 115
In this system, both of these symbols would be acceptable ways of writing such numbers.

Now we get to the sociolinguistic or 'socionumeristic' use of these numbers: of course, pricetags with repeated digits are preferred - ßßß rather than 1332. Reducing the number of digits, if possible, is also preferred: ß999 rather than 12999. Stating a debt using Ð, Þ, and ß - if you're the lender - is perceived as a kind way, an indication that there is no hurry. Stating it using the canonical western form indicates some demand of payback soon.

Now, for a mathematical challenge: give an algorithm f(N), which as an output gives the number X, 0 < X ≤ N, such that X has the maximum number of ways to be written using this notation out of all integers less than or equal to N.

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