Mystery of the misprinted dice

Question

Dan Hedfelt is a small gaming equipment manufacturer.

Recently, he upgraded the machine that makes six sided dice, but the new machine started acting up, and he faced a production quality issue: the machine worked otherwise perfectly, but for about once in every ten dice, it somehow swapped every side's print to the exact opposite side.

Dan was initially worried about this, but since the error left the resulting dice perfectly fair and usable, he just handed them out as small gifts for his customers; he was actually feeling a kind of affinity with those misprinted dice.

After signing the contract to sell the old machine, he then headed to the tennis club, since his twin brother had once again challenged him to a rematch. Once there, his brother noticed a smudge on Dan's hand. "Oh yeah, that happens a lot", Dan replied. He then promptly proceeded to defeat his brother, as was usual.

So finally, here’s the question: Dan was somehow able to tell the misprinted dice from the regular ones. How?

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This puzzle is a kind of a multipart hybrid deduction riddle, where almost everything in the story is a clue of some sort, and the final answer should fit them all.

(The accepted answer is for an earlier version of the puzzle, which it nicely solved. The new and improved version was also solved in a different answer, which didn’t repeat the old answer’s findings. The complete solution to the current version is available in the community wikified answer below. I apologize for the inconvenience.)

Answer 1

I think

Dan is left-handed. Left-handed people often get smudge on their hand when writing something. Dan has just signed contract, so he got ink smudge on his left hand. Because it was before tennis match, his brother noticed that smudge. Plus, it doesn't mean Dan's brother also left-handed because they are twins. Even though they are twins, there are reasonable chance that dominating hand is different between twins.

Answer 2

I think the answer is:

Dan is left-handed. Conventionally, on a normal western die the faces of the sides labeled as 1, 2, and 3 share a corner and are right-handed (1->2->3 appear in a counter-clockwise ordering), but when you swap the opposite-faces of the die they become mirrored and are left-handed (1->2->3 appear in a clockwise ordering).

However...

Chinese dice are conventionally left-handed, so he doesn't need to give them away, he can open up a whole new market in China!

Answer 3

Answer was taken from Kamome, just adding 3 of the clues from the story here since you can't use spoilers in comments.

"but for about once in every ten dice, it somehow swapped every side's print to the exact opposite side."

The fact that one in every 10 people is left handed

Dan wins at tennis, as per usual

Referring to the fact that left handed people, or lefties, have an advantage in tennis.

And maybe replacing the old with a new machine refers to

The fact that in old school systems, children were taught to write with their right hand, even if they were left handed. Nowadays children are free to write with their dominant hand.

Answer 4

Since the answer came in many parts, and from many people (and I kinda messed up with the question when I originally posed the question), it's a bit hard to get all the bits scattered around in the various answers and comments.. Therefore, here is:

The Official Spoiler

The answer in a single word, is

Left-handed

Here are all the intended clues, piece by piece:

Dan Hedfelt is a small gaming equipment manufacturer.

"Dan Hedfelt" is, of course, an anagram of "left handed"

Recently, he upgraded the machine that makes six sided dice, but the new machine started acting up, and he faced a production quality issue: the machine worked otherwise perfectly, but for about once in every ten dice, it somehow swapped every side's print to the exact opposite side.

This is the geometry bit. Swapping two opposite sides of a dice changes its chirality, meaning that that the dice will become a mirror image of itself. Swapping all the sides is equal to three such swaps. Any two swaps cancel each other, so the result is a mirror image of the regular dice.
Apart from geometry, the 1-in-10 refers to the approximate ratio of left-handed people in the human population.

Dan was initially worried about this, but since the error left the resulting dice perfectly fair and usable, he just handed them out as small gifts for his customers;

Reading only the verbs in this sentence, you get "was" "left" "handed". (Yeah, I wasn't surprised nobody got this..)

he was actually feeling a kind of affinity with those misprinted dice.

The nature of this affinity is the crux of the matter, and will be resolved shortly..

After signing the contract to sell the old machine, he then headed to the tennis club, since his twin brother had once again challenged him to a rematch. Once there, his brother noticed a smudge on Dan's hand. "Oh yeah, that happens a lot", Dan replied. He then promptly proceeded to defeat his brother, as was usual.

These are all hints about Dan's left-handedness.
- Statistically, it's about twice as likely that a person is left-handed, if he has a twin.
- Left handed people writing with a pen do often get smudges on the hard-to-see part of their hand, since when writing from left to right, the left hand tends to touch the paper at the very spot where the wet ink is. Apparently this happened when he was signing the contract.
- When practicing tennis, the vast majority of your opponents will be right handed. (This is true, even though lefties are somewhat overrepresented in tennis: 15 of the ATP top 100 are southpaws.) Therefore, when playing against his twin (who is right-handed; twins often prefer different hands), Dan has the advantage of having practised more against similar opponents.

So finally, here’s the question: Dan was somehow able to tell the misprinted dice from the regular ones. How?

The geometric term "chirality" actually is just a synonym for "handedness". The regular western dice are said to be "right-handed", meaning that when you look at the 1-2-3 corner, the numbers run in the counterclockwise direction. The misprinted dice, having their chirality reversed an odd number of times, are "left-handed", meaning that the numbers 1-2-3 run in the clockwise direction.

So, the reason for the affinity was that

Dan was left-handed, and so were the misprinted dice.

Shouts out to Kamome, who solved the original puzzle without needing most of the clues and got the tick for his efforts, RnRoger, who figured out the majority of the remaining clues, Foo Barrigno who finally connected all the clues to the title, solving the newly edited question, and M Oehm, who was the first to figure out the anagram.

Showering each of them with upvotes wouldn't be a bad idea at all.

Thank you for your interest!

Answer 5

After swapping printings,

the triple 1-2-3, which is usually right-oriented, would still be right-oriented when seen from its original position through a transparent die.

However, as dice are usually not transparent,

we need to look from the opposite side to see 1-2-3, and then the orientation is swapped. It's like clockwise and counter-clockwise when watched from opposite sides of a window's glass (edit: of the same window's glass).