Problems without solutions

[Liquid]

Somthing I considered earlier. Has anyone come across a problem that needs improvement?

For example my friend saw a problem in the security of RFID cards without a solution and solved it.(patent pending)

I myself solved the issue of high transport costs associated with washing machines due to the concrete blocks(along with the inevitable back bad that comes with moving them(patent pending)

However it makes me wonder how many issues out there plague people go without a solution. Ever come across one yourselves?

[Anonymous]

But that would entail those men leaving the pub with £1.67 each in their pockets - which they clearly didn't.

[Liquid]

I'm confused now my maths was the same as steersy I just ignore the 1p difference.

•If the total bill was £30 between 3 men each giving £10.

•The manager then decides to give them a discount of £5 after the 3 men have already given their money to the waitress bringing the total bill to £25.

• They split the £5 4 ways giving each male £1 and a £2 tip to the waitress.

•the £5 is ignored as it is the discount its surplus to the problem at hand.(red herring)

•£25 bill is paid.

•£5.00 surplus is split between the 3 men(£1 each) and the waitress (£2)

£25.00 bill (£8.333 recurring)
£3.00 discount (£1 to each man)
£2.00 tip (assuming equal share each man gave £0.666666 recurring)
(£30.00) total.

Each man £8.33 bill + £0.67tip = £9.00 they each leave with £1 change from a £10 entrance worth.

Maybe I'm over thinking this:(

[Anonymous]

As a maths "problem" it's simple enough - the "trick" is that wording it allows you to flip the sign on the values - effectively adding negative values by jumping between the £30 and the £25 then playing with adding or subtracting the other £3 and £2.

It's a similat "trick" to the one where you "prove" that you have 11 fingers - by counting backwards on one hand - ticking the fingers off as "10, 9, 8, 7, 6" and then holding up the other hand and saying "...and five makes eleven".

It's not a "problem without a solution" but it's a good illustration of how a "problem" can appear unresolvable or paradoxical simply because the question itself is wrong.

The classic "unresolvable question" is "What happens if an irresistable force meets an immovable object?".   It seems like there must be a solution - probably something involving a big bang - but in reality, it's a "trick question" because the two terms are mutually exclusive.  Within any universe, you can have one or the other - not both.  For an irreistable force to exist, there can not be an immovable object and vice versa.

"Problems" like that are not posed with any intention of finding a solution - their value is in training the mind to work a certain way.  It's like a footballer doing press-ups - of no specific benefit to the game of football but useful for increasing stength and stamina in general.

In reality, there are no "problems without solutions" - just problems that haven't been understood yet - or problems that can't exist until another one has been resolved.  You don't ask the question "how do we make a spacesuit?" until after you've discovered that one is necessary, for example.  Until you know that space is a vacuum with temperatures approaching zero Kelvin, you wouldn't even know that the question of special clothing existed.  

Problems are evolutionary - they arise in sequence and only after an earlier question has been posed and answered.   What often happens - like the "three men in a bar" problem - is that the problem is not understood well enough to be able to ask it in a way that the solution is apparent.  

It's the measure of genius when someone is able to see supposedly complex problems in simple terms.   That's why people who really understand something can explain it in simple, plain English rather than using jargon and techno-babble - they understand "why" rather than just "how".

I had a friend at school who could "visualise" maths problems - he didn't see the squiggles on the page and work them out step by step - he saw them as physical things that he could move around inside his head - they way we counted on our fingers as kids to make addition and subtraction "real".   He said he'd always seen maths that way - had no idea that the rest of us didn't.  Thing is, he often knew the answer to a problem but couldn't explain the math on paper - so he had a problem because you have to provide proof to get marks and pass exams.

[perksie]

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@Liquid wrote:

Maybe I'm over thinking this:(

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Keep it simple:

They paid £30, they were returned £5, £2 of which they gave away.

They spent £25 and walked away with £3, which is what they paid less £2 to the waitress.

The mistake is in assuming they paid £27 which they didn't. QED.

[Anonymous]

Not the case - they really did pay £27 between them - each gave £10 and each received £1 back.

The whole thing is based around a semantic fallacy - it's actually not a math's problem but a mathematical mistake - which is why there is no "solution" as such - just an explanation for what the riddle says that is wrong.

Bill + Change + Tip = Paid
25 + 3 + 2 = 30

..so we can say that...

Paid - Change - Tip = Bill = 25
30 - 3 - 2 = 25

..because any number "crossing the equal sign" is flipped between positive and negative in order to keep the sum balanced.

What the wording of the of riddle is saying is...

Paid "MINUS" Change "PLUS" Tip equals 29
30 - 3 + 2 = 29

...which introduces two separate errors.

That £2 tip should have been subtracted AND the value we needed the math to resolve to was the bill (£25) - not the amount paid (£30 - which had already been included in the equation).  In fact, the bill - £25 - is now completely missing from the equation.

The wording of the riddle has added the tip when it needed to be subtracted AND it's then deliberately changed the value you were actually looking for - swapping the £25 "bill" for the £30 "paid" - to hide the trick - making it much harder to spot that the "error" is in handling of that £2 tip.

This sort of error does occur in the real world.  It's very easy to see how someone could include something like it on their accounts or expenses claim because words are not bound by any logic - as long as each sentence makes sense, it's easy to lose track of overall sense of the situation.   You can put pen to paper and ask "What's the difference between a frog?" - and it read's just fine - but the question has no solution because it contains a "mistake" that is not immediately apparent.  It's only when you apply the external knowledge - that "difference between" is a comparative and therefore requires a second subject - that you can see why a seemingly perfectly well written question can't be answered.

[perksie]

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@Anonymous wrote:

Not the case - they really did pay £27 between them - each gave £10 and each received £1 back.

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It is true, they did not pay £27 at any time, they didn't get £1 back they got £5 back.

Maths can be re-jigged to say almost anything, but the fact is they paid £25 received £3 back and went away with £3, £2 went to the waitress and their bill was £25, however else you say it it's wrong.

[sheepdog]

I propose that PengiPete donates the missing £1 to the forum Christmas party 

Sheepdog in "BODMAS" mode

[Anonymous]

"they did not pay £27 at any time, they didn't get £1 back they got £5 back."

Not so - they really spend £27 and they really did get back £1 each - they really did end-up spending £9 each for their lunch break.

Once you know the answer, it's obvious and incredibly simple - the tip gets counted twice.

The diners handed over £30 and received £3 back - meaning that they spent £27 - £25 for the food and £2 for the tip.  The really did spend £9 each.

So the £27 total that they spent actually includes that £2 tip - but that same £2 is then added to the £27 creating the £29 - which is then compared with the original £30.

It has nothing to do with the unit cost of a plate of food or anything else complicated - just a really simple trick with words that make you forget that the tip was already paid before they received their £3 change.

As a puzzle, it one of those where you know it's "wrong" because it has to be - the hard part is saying how it was done.

There is a real-world scam called "Change Raising" that uses the same principle.

[Anonymous]

£29.99 sounds like a cheap O2 pay as you go phone