Return to the Island of Women
On the sixteenth night, however, a number of men were executed, and on the seventeenth day the faithful men of the island awoke with their wives to see the judgment.
How many men of the island were adulterers? How did their wives find out?
Before reading this post, please review the riddle in The Island of Women.
Let’s Talk Strategy
Given an extremely challenging problem like this one, it’s generally a terrible idea to plow into attempting a solution without first taking a step back to consider strategy. So let’s pause here to consider what problem-solving tools we can bring to bear on this riddle.
- 1. When faced with a difficult problem, especially one involving numbers too large to consider directly, it is useful to begin by solving a simpler analogous problem instead.
- 2. Understand that every single piece of given information must have a purpose. Keep track of what you’ve used and what you haven’t, and then figure out how the unused information matters.
- 3. Different fields call for different tools. Try to determine what’s really being tested, and then bring subject matter-specific tricks and strategies to bear on the problem.
- 4. If a straightforward approach to a difficult problem fails, a shift in perspective can often help; don’t forget to ask the question, “How else could one look at this problem?”
- 5. Have faith — both in the basic solvability of the problem and in your ability to solve the problem using techniques within your grasp.
Strategy in Action
Now we’ll apply each of the above problem-solving tools to the challenge at hand.
- 1. Solve a Simpler Problem: This riddle involves consideration of a sixteen-day-long sequence of events. That’s hard. It demands a solution based on an unknown number of murdered adulterous men. That’s hard too.
- Let’s make it easier. What if exactly one man were a cheater? There must be at least one — the queen observed unfaithfulness — so this is the simplest possible version of our problem. By figuring out what happens in this version of the problem, we’ll gain important insights into the nature of the more-general solution.
- 2. Understand Every Piece of Information: A well-crafted puzzle does not come with extra pieces. Any given bit of information must be useful — either as part of the solution or else as a trap designed to lead us astray based on some specific logical fallacy.
- So what’s the deal with the whole sixteen days thing? Nothing seems to be happening, right? How is it that night sixteen is different from any other night? Something must be changing. But apparently the only new information each day is the passage of another murder-free night. Could the passage of a murder-free night itself provide useful new information?
- 3. Use Subject-Specific Tools: Did you notice the hint? It’s subtle, and it’s not where you might expect to find it. Look at the tags — of both this post and the riddle post that came before: apropos of nothing, logic puzzles, inductive reasoning.
- Inductive reasoning? That’s the logical/philosophical idea of working from specific cases to formulate a general rule. Inductive reasoning is pattern-driven. We should examine a few easy specific cases and then try to generalize a rule.
- Let’s think about what specific cases might be useful starting points for this riddle. Once again we come around to the one-cheater case. We really ought to start with that.
- 4. Shift Perspective: If a problem involves a given area of study, consider how a different area of study might approach it. If a problem seems to be all about a given formula, consider what other formulas might prove surprisingly useful. If a problem implicates multiple parties, consider how it might look from the other parties’ perspectives.
- The first two ideas, involving different areas and different formulas, don’t do much for us in this case. But if you’re grinding away looking at the one-cheater picture and getting nowhere, imagine what someone else might see. Specifically, think about what a woman on the island sees:
- A. If she’s not the one with the cheating husband, she sees pretty much what you see: a bunch of faithful men and one cheater. (The only difference compared to your perspective: she can’t evaluate her own husband and see that he is faithful.)
- B. If she is the one with the cheating husband, though, what she sees is very different: nothing but faithful men, the whole island over. (But again, with a blind spot for her own husband!)
- 5. Have Faith: This riddle is extremely challenging, and it’s virtually impossible to see through the whole solution from beginning to end. Thus it’s especially critical in this case to trust your ability, the utility of the logical tools at your disposal, and the basic logical solvability of the task at hand. Just as a good puzzle does not come with extra pieces, a good puzzle does not come with missing pieces either. A certain amount of faith is required.
- Furthermore, a very difficult problem doesn’t generally involve pieces that fit together in bizarre new ways. It just had more pieces and makes it more difficult to spot the pieces that fit together. You still have all the tools you need to solve the problem.
- How does this point apply to this riddle? Simple: just find a path that makes some sense, even accepting that you cannot see where it goes or how it might lead to the ultimate solution. And then walk that path. Put one foot in front of the other. Don’t focus on the solution, but don’t give up, either. Just take the next step — and know that the next step is always in reach.
- Who knows how the “one cheater” case will help us. It certainly can’t hurt to take the step and find out. Now let’s see where it goes…
Solving the Riddle
We’re finally ready to put it all together and walk through a full solution to the riddle.
Begin with the one-cheater case, and think once again about what the various women would see. Most women — all but one, actually — would see a single cheating man on the island. One woman, however — the one with the cheating husband — would see none. But the queen said that there were cheating men on the island. This woman could reach only one conclusion: her husband must be the cheater. So she would murder her husband, right on that very first night.
What if there were instead two cheating men? Most women would see both, but two women (those married to the cheaters) would see only one apiece. Well, what if you see just one cheating man? What do you expect (and dearly hope) will happen on that first night? A murder, of course! Because if there really were only one cheater, then almost every other woman would see one cheater, one woman would see none, and that woman would kill the cheating man on the first night. See, the passage of a murder-free night really does provide useful new information. It tells the women that there are more cheating men.
So if the first night passes murder-free, you know that there could not possibly have been just one cheating man, or he’d already be dead. So there must be two. But if you see only one, well, that’s bad news for your husband, because you’re going to kill him in his sleep on the second night.
What if there were three cheating men? The analysis is old-hat by now. Most women would see three cheaters, but three women would see just two cheaters apiece. Those three would anxiously await the second night, hoping for murders. But when the third day broke with no dead adulterers the only possible conclusion would be that more than two cheating men were to be found on the island — and each of the women who saw but two would set about planning her own husband’s imminent demise.
If there is one cheater, he dies on the first night. If there are two, they die on night two. If three cheaters are present, they survive until night three.
So we come at last to the sixteenth day. Had there been exactly fifteen cheaters, the fifteenth night should have featured fifteen untimely deaths. In fact, though, nobody wakes up dead on this sixteenth morning. Sixteen unhappy women each see but fifteen very-much-alive cheating men, and each of those sixteen women reasons out that a sixteenth cheating man must be hidden from her view. Sixteen unhappy women begin quietly planning sixteen grisly murders.
On the sixteenth night, sixteen cheating men are killed. And the power of logic is how their wives found out.
Tags: Apropos of Nothing, Inductive Reasoning, Logic Puzzles
This solution probably is a better example of how assumptions can make an ass out of u and me. The premise of the solution quite simply doesn’t work. If there are 16 adulterers then on the first day all the women see either 15 or 16 adulterers, not 0 or 1, so having no one killed on the first day, or any subsequent days, doesn’t add any more information.
Graeme,
I want to apologize for the delay in posting your comment and in my reply. We’ve been getting a lot of spam, and your note was temporarily lost in the shuffle.
Now, on to your comments:
Harsh tone aside, you’re exactly right that the women don’t see 0 or 1 adulterers on day 1. However, you’re wrong about the implications of that fact and about the ultimate validity of the solution.
The information that the women gain on each subsequent day is the simple fact that no men have been killed (and this is why the women must all look upon all of the men of the island every day). Each day this tells the women something new. On the second day, the fact that no men were killed tells all of the women that there wasn’t exactly one cheating man (as a solitary cheating man would have been killed on the first night). On the third day, the women learn that there weren’t exactly two cheating men (as each wife would have seen only one cheating man, but would have known there must be more than one after the lack of murder on the prior night). And so forth.
In our case, there are truly 16 cheating men. Each wife of a cheater sees 15 cheating men. And each such wife therefore *expects* those men to die on the 15th night. When those cheaters survive night 15, the only possible explanation is that there were more than 15 cheaters. There were 16. And the 16th must be that woman’s husband.
The idea here is to extrapolate from the case of 1 cheater — NOT to claim that there is truly only one such man. We’re just trying small cases to see a pattern. Again, imagine being the wife of a cheater in the “two cheaters” case. Imagine seeing only one cheating man. Imagine what you would think when that one man didn’t die after night one. What would you have learned? Now try it again with three cheating men. What if one of those three is your husband? What would you see? What would you expect to happen based on the “two cheaters” case? Two dead men after night two, right? But now imagine what you would learn when the two cheaters you could see lived through night two. And so forth.
I hope some of this helps. Inductive reasoning is difficult, and this puzzle is very hard, but I assure you the solution works. I’m happy to discuss it further if you’d like.
–Anthony Ritz