Brain Teasers are Back!

A more obvious and therefore more convincing solution is 7 & 4.

But I haven’t worked out any significance of Sam’s final text other than the literal meaning.

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IB,

You and Dozey have both correctly identified 11 as a possible sum, that is being consistent with Sam’s first statement that Pru can’t solve it.

It is also consistent with Pru being able to solve it since Pru knows the possible sums (even though Dozey, Don and me haven’t aligned on them yet!). Dozey has correctly identified that 17 is a possible sum, so we can rule out 30 as a product as it could give 5x6 or 2x15.

For Sam to make his final text, he needs to distinguish between 2x9, 3x8 and 4x7 for one of them to be the unique answer.

You, Dozey and I think Don seem to be tantalisingly close!

Well, maybe it is 3 and 8 then. The product can be generated by multiplying two numbers, neither of which are primes?

My recollection (it’s some time back) was that the original “teaser” allowed Sums up to 100 (but only with no integers repeated). The originator thought that there was only one, unique pair of numbers that could fit the text of the teaser. But I think he was wrong, and there was another pair.

By carefully examining all versions of the “Teaser” with pairs of numbers Summing progressively from 3 up to 100 it was found that some such upper-boundary “pairs” eg 20, 30, 40 did indeed have a solution that was limited to a single, unique pair. Other pairs, eg 50, 60 ,70, 80 , 90 and 100 had more than one solution - usually two solutions and occasionally three solutions. The other boundary pairs in between eg 51, 64, 82 occasionally had a unique solutions but many had twin or triple solutions.

Dozey and IB seemed reluctant to go through the tedious process involved in eliminating the “easy solutions” eg the primes and the unique pairs eg 2x4=8 etc already discussed. Based on my memory, I recalled that 20 would give a unique solution with a significant reduction in the processing time, and what’s more, my recollection was, that it would generate exactly the same pair of numbers in the solution as the upper-boundary 40 would.

But I might be wrong. Especially since, as I said above, the version that I worked out a few years back, didn’t allow the two numbers to be the same. So if there is any doubt, stick with 40.

I think I’m done, so I shall explain my reasoning, whether or not there are other reasonings with other answers (my earlier 2 & 9 being one, but more obsure than the obvious one that I describe below):

Sam’s message has 11 spaces (it couldn’t be the number of letters, characters or characters+spaces as all are >40).

Pru looking at that deduces the same. Because to write a text with the number of spaces giving her number, she instead did the maths herself before responding considering which two numbers with a sum of 11 multiplied together would give her number. 29, 38, 47 or 56, and her response gave one of those two numbers to Sam, i.e. 4. Clearly from her response the other number must be 7 and her number 28.

Sam on receiving the text eventually realises what Pru had done, subtracts 4 from 11 getting 7, and sent Pru a straight, non-coded text confirming he’d got it.

Well, that’s about as valid an answer as I’ve ever seen.

I don’t know if Ravvie will agree, but it would be difficult not to.

I certainly have two numbers in mind and they aren’t the same as yours, but hey, it’s generally accepted that what people say is important, and that applies to this teaser, more so than many !

Are Sam and Pru in competition and trying to mis-direct one another. Or are they trying to help one another. Or are they unwittingly providing useful information to each other … or … ?

Well, IB’s coded message solution is the most lateral thinking that I have come across, so well done! Not much chance of me topping that.

It is sometimes possible to come up with a different unique solution, depending on any assumptions being made. It’s the arguments given in the solution that’s important. Whilst I had tried to make the problem as unambiguous as possible, clearly I didn’t consider the coded message approach.

The intention was to assume nothing more than was given. That is, they are given a problem to solve and state the truth. So no misdirection between them was intended. Being best students I intended it to mean they don’t make mistakes.

I plan to set out my solution tomorrow, giving Don and anyone else a chance to finalise their solution first. I will include a table along the lines Don suggested assuming I can work out how to include it on this platform.

Ok, I’m following two lines of investigation.

The first and foremost is to track-down, sort and file my various teasers, scribbles and other bits and pieces, to find this teaser from a few years back. It wasn’t on this forum and it might be a slight (hence significant) variant from that posted above. ISTR there were a few variations in the upper bounds of the unknown Sum and whether the two numbers that formed that Sum had to be different or could be the same.

The second is to re-work the teaser from scratch. This has led, so far, to three quite different solutions, not at the same time, but following separate attempts, depending on my thought process and methodology.

In the latest version I have whittled down the possible Sums held by Sam to be 11, 17, 23, 27, 29, 35, 37 and possibly 41.

It then looks as if 17 is the most likely candidate comprising 4 and 13.

However, two previous deliberations resulted in Sums of 8 (2 + 6) then 11 (2 + 9)

So, a bit more work on these fronts to see where and why I keep diverging.

But I rather think that a search of the loft might be more profitable. :sunglasses:

Don,

I get 2,6 as a possible solution but I haven’t thought how to show it is unique (and it may not be). To do this one needs to assume that Pru or Sam would have contacted each other if they had worked out the answer immediately. I must say I had to do some research online to find this one.

If we make no such assumption then I get 4,13 as a unique solution. In this scenario nothing can be inferred by the other’s silence ahead of the correspondence. I will post my solution later on this variation unless you want me to hold back whilst you sort your loft!

I am now feeling a bit guilty as I have been putting off fixing a broken cabinet in the bathroom! Maybe that’s why I haven’t found the arithmetic tedious - I would do anything to put off routine DIY!

I have had a flooded basement to contend with, which has been a bit distracting!

A bit of a distraction. Nice understatement !

Hope you have it under control and manage to get it sorted very soon.

Well, I found a few of my old notes, but they are somewhat faded and difficult to read. I will persevere.

First will be to confirm that the storyline is, in effect, the same. Eg the participants in my notes spoke by telephone … what ? I hear you say. A “telephone ?” What on earth is one of those ?

Ok, I haven’t been able to fully decipher my old notes. However, after re-examining my recent workings and also confirming to myself that the teaser posted by Ravvie is basically the same teaser that I had from some years back, I feel confident enough to go with my initial logic and recollection of 6 and 2. (I have put it this way round for a reason !)

I found, to my dismay, that when re-working this teaser from scratch, that I kept getting distracted and kept mis-interpreting what each of S and P could actually know. So it’s taken quite some time to set things out logically.

I am trying to write my logic out at the moment and it’s difficult enough. But proving even more difficult with VE Day observations taking place.

So, I am going to mark my place with 6 and 2 and hope that when I get back to finish off writing up my logic, I don’t find a bloomer.

Don

Great. I think a subtle difference in the interpretation leads to two quite different solutions. I expect both to be valid, unique in their own right being consistent with one’s underlying assumptions.

Would you like to write up the rationale for 6,2? I can do 4,13

Sorry, this one has blown my head.

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Oh dear! Maybe I overdid it a bit with this one, what with my bathroom cabinet falling off the wall, Don taking refuge in his loft and worst of all, Dozey contending with flooding in his basement.

I will post my solution, hopefully it won’t look too daunting, but probably not a good idea for me to try it tonight as I have had a couple of VE gin and tonics.

You’re not alone Mike !!!

I have started setting out my logic, and even though I know how I solved this a few years ago, actually writing the logic down … is taking a good bit of effort.

At the moment it starts …
…If Pru’s Product is unique, she would have factorized it and identified m and n immediately (m and n are the two elusive numbers, which are not necessarily different). She didn’t, so the Product of m and n can’t be unique and therefore must be the product of at least 2 distinct pairs of valid numbers, AND …
Sam’s statement MUST convey some info that makes it possible for Pru to select the correct pair from amongst those two (or more) different candidates.
That first call from Sam to Pru is key.

I have got a bit further. I simply provide the above “snippet” as a taster of the headache that lies ahead !! both for me to write, and others to read (if they are so inclined !!)

Perhaps you could mull over it this beautiful sunny Saturday morning whilst you lounge in your garden, beer in one hand, steak sandwich in the other, gazing out to sea … :sunglasses:

Well, that’s a plan. The sun is out in our garden by the harbour. I’ve been to the supermarket and have steak and beer for tonight!

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Errrr Mike …

Don’t forget the mulling over bit :sunglasses:

Darn, I thought you may have missed how I excluded that bit…

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