bonanova

I'm taking a slightly stricter view, that NEWS coding is not available. Wolfgang has allowed that prisoners can either face toward or away from the door. That is, NS (or EW, as I drew it) coding is available, and that requires two prisoners to signal a color. That presents a particular problem for P2 on initial entry, but I think it can be worked around.

Wow. I apologize for starting an identical[!] puzzle in the logic forum. Three days apart. What are the odds.

For a score of 42.5%. Not bad!

Agree that now it's not difficult. But the first two might need change row locations. Suppose the first three prisoners were [1] all red, [2] all different colors.

The given formula 1 + 6 x 2(N-1) indeed discloses the order of magnitude, a reasonable approximation, and an upper bound. Inspecting its divergence from the actual count [i could not compute recursively past N=30 in reasonable time] disclosed: the correction begins with the result from the seventh-preceding result. It increases from that value going forward, exponentially, and always a factor 6, but I could not find a closed form. So a closed form for the total answer remains to be found. But execution efficiency for table look-up was dramatically better: Whereas recursion prohibited going past N ~ 25 in reasonable time [a dinner break], N= 100,000 with table look-up took only 20 seconds. Convergence to the 2/7 value to 10 decimal places was achieved for N ~ 75.

It is a beginner-level bit of mathematical magic to prove that 1=2. This demonstration probably occurs multiple times in the history of this forum. Its fallacy depends on embedding, and cleverly hiding somewhere in the "analysis," a division by zero. Which of course is not permitted. So regarding this matter our world is still a safe abode. Nonetheless, this equality still has a life for puzzle solvers. By the application of a familiar proof, the addition of a single well-known mathematical symbol, and without resorting to prohibited mathematical operations, 1=2 can still be shown to be true. Have fun, and please use spoilers.

Continuing with the community solve approach, can we clarify: Must all the persons in a particular row face the same direction? If prisoners in say the Yellow row can face in differing directions does that constitute [illegal] communication? Can the first two prisoners choose the time that they [legally] alter their position, at any time up to the last prisoner takes his place? During making raws...some of them can be..back to back (in any raw)...but by reaching the last prisoner,they all shoud be at the same direction But only the first two prisoners can move. This means the seventh prisoner, say can come into a row back to back with someone else in that row, then at a later time switch his direction so that at the end all in that row have the same direction. So as long as a prisoner stays in the same position, he can flip his direction back and forth during the formation of rows. That sounds like communication. Another piece of jello fell off the nail...

I got a similar order; my table of numbers differs though. The given closed form shows an understandable accounting of the required executions, as N increases from zero. The execution count includes: 1. One, for the initial execution, which directly returns 1, if N is zero 2. Additional executions, in Sets of six, comprising executions of the entire formula, when N is not zero. 3. The doubling of the number of Sets, as N increments, owing to recalculation of previous results. 4. The total, which is simply 1 plus 6 times the partial sum of the powers of 2. Are you asking for a different closed form; one that gathers multiply-calculated results into 137 groups, then sums those numbers? Perhaps making those tables for a few smaller values of N will suggest the form of that expression. Programming question: could execution time be drastically reduced, by storing previously calculated values of P and calling them from a table if needed again?

Yes...the raws should be as you mentioned in your diagram,but all should face one direction "All" in one line? Or "All" in the whole room? Say the the persons in the red line face the door. Can the persons in the green line face away from the door? This is like nailing jello to a tree.

Another point to clarify. Must the rows be parallel, and in particular Y-G-R-B order? OP seems to suggest this condition. +-------------------------+ | | | Y Y Y Y Y . . . | | G G G . . . . . | Door R R R R R R . . | | B B B B B . . . | | | | | +-------------------------+ Or could the rows e.g. all start at the middle of the room, and grow outward toward the four walls, in any sequence? Say Yellow to the north, Green to the south, Red to the east and Blue to the west?

Hi sergyegi, and welcome to the Den.

Prime, your analysis lays the groundwork for a great trivia question: If I repeatedly roll a fair six-sided die, what number has the greatest probability of occurring in the sequence of running totals?

Place letters in a 3x3 grid in a way that permits spelling the greatest number of three-letter English words. The same letter can be used twice in a word. Letters adjacent in the word must touch, in any of eight directions, on the grid Example: T R E Y D Z U N G Permits [among possibly other words] ere try gnu red zed dun dry ... Entries must contain a minimum of ten words.

Hey araver, LTNS!

Continuing with the community solve approach, can we clarify: Must all the persons in a particular row face the same direction? If prisoners in say the Yellow row can face in differing directions does that constitute [illegal] communication? Can the first two prisoners choose the time that they [legally] alter their position, at any time up to the last prisoner takes his place?

Yes...thats right....he should wait until the previous prisoner is standing in his raw and took his position. Nice approach. I agree tho that the warden might object. Also, wouldn't the nineteenth prisoner also need to execute this [seemingly one-time-only] ploy?

Wolfgang, I think the point that asks for clarification is this: Must each prisoner wait, outside the room and blindfolded, until previous prisoners have taken their positions?

Chess puzzle thread is a reasonable suggestion. Solver can be the one to post a new puzzle.

To me it seems that 'Entering singly and not changing one's position once taken" are conditions with no effect, if "hanging out" is permitted. What relevance to the solution can that guidance have, other than to prohibit that option? Yet that is what adds the difficulty that I see here. If language issues have made the point debatable, Wolfgang will have to clarify.

Bushindo has seven decimal places; the Captain has it exactly, if surprisingly; and Prime has the insight. A triple tag team solution! Good job all. So who gets the "solved" tag .... ? The nod goes to Prime's explanation.

This is a challenge. A two-color version requires and permits movement after taking one's position. So I don't think it provides useful guidance for this case Great puzzle. Eager to see the solution.