It seems strange to me that something that means half can also mean odd. Since if you can divide something in half it means you were starting with something even. Also dividing something in half does not mean it’s necessarily odd either for instance half of 20 is 10 which is still even.
Anyone on familiar with the etymology of the kanji? The reason is odd and half just seem contradictory to me.
It seems to be specifically when referring to dice in a game called 丁半 (ちょうはん). I can’t find evidence that the association exists outside of that game.
I guess it’s not impossible for it to be extended outside of that, but if you want to talk about odd numbers in standard math terminology, you should use 奇数 (and 偶数 for even numbers).
That’s was the context I saw it being used for odd. Mainly, this “丁か半か” → Even or Odd
When looking at the Chinese etymology, it mentions just even/half. So how did Japanese also associate odd with this? If it’s just from this dice game. Why did the game decide using that for odd?
It seems like you’re assuming the meanings of odd and even came first and then got applied to the game. It’s also possible the even and odd associations came with the game as the origin.
According to this page 丁 is a shortcut for 度丁 and 半 for 半端.
So maybe it’s because that’s what happen when you divide by 2 ? For even number it’s 丁度 (exact, just right), but for odd number it’s 半端 (remnant, fraction, imcomplete)
Ah, that makes more sense. So there using 半 as more of meaning parts which would make sense, since cutting something in half leaves you with 2 parts.
Seems to be it, more details here:
「丁」は偶数で半分に丁度割れる数なので「丁度」の「丁」、「半」は奇数だから割り切れないので「半端」の「半」だそうです。 つまり、「丁度」「半端」を略して言っているんだそうです。
半分に丁度割れる or 割り切れないので「半端」
A comment: this game gets portrayed quite a bit in samurai anime. The impact of gambling on the lives of people back then is probably well-researched as well. Such as fathers gambling away their money, and daughters being sold into prostitution to make up for the debt. (that’s one theme you’ll see in Samurai Champloo for example, but I’ve seen it elsewhere also (can’t remember where)).
The Pachinko culture is strong with Japan as well, though it’s not deemed gambling since you can’t earn money, just prizes from it. For some reason I’ve always thought this semi-gambling might have been a reaction to the older gambling games such as 丁半 or 麻雀 (マージャン) - as a safer venue for the same urges to bet money. 
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