Combinations - most hateful of most maths...

[lau_kai]

This is an utmost annoying question that i assume requires lots of patience(for people like me...)... or just a few secs for mathematicians...

Joe asked Penny to think of four different postive digits and add up the 24 different 4-digit numbers that could be made using the four digits. Then he asked her to subtract just one of the 4-digits numbers from the total and write down the new total. When asked what number she had written down, Penny replied: "122**0".

Joe didnt quite hear the 4th or 5th digits. But he was able to work out the number Penny had subtracted. What was that number?

well, is there any combinations of formluae that may work on this question? or can it be onli pure guess and check?

Thanx in advance

[jemidiah]

Very interesting problem...

[musings comment=They turn into good stuff, I promise]
I think that taking the problem in averages would help here.

There are 24 different numbers, each of 4 digits, though only 4 actually different single digit numbers per digit of each of the 4-digit numbers. Digit. For the sum of these 24 different numbers, we can take each digit of the number, average it, multiply it by 10ⁿ depending on its placement in the original number, and add them all together. Aka, here:

Let's choose three different single-digit numbers, a; b; c; and d.

In the 26 possible combinations of those 4 digits, each one will repeat at each decimal place 24/4 or 6 times. a*1000 + b*100 + c*10 + d*1, b*1000 + a*100 + c*10 + d*1, etc.

This can then be simplified to 6*a*1000 + 6*b*1000 + 6*c*1000 + 6*d*1000 + 6*a*100 + 6*b*100 + ...

This can then, again, of course, obviously, and to avoid putting us in a coma (sorry for the bad pun ) be simplified to 6*([a+b+c+d]*1000 + [a+b+c+d]*100 + [a+b+c+d]*10 + [a+b+c+d]*1).

This can then, again, yet of course, be simplified to [a+b+c+d]*(6666), meaning that the original number was divisible by 6666 (the sum of the 26, before another number was subtracted from it).


So, what was the original sum? Logically, it has to be greater than 122,000. Also, the very smallest number from four different single-digits stuck together subtracted from it would be 1234, and the very largest one would be 9876, So, we're looking between 122,000+1234 and 122,000+9876, inclusive. Ok, I really hope that there is only one multiple of 6666 in that range: yay, there's only one. Who'da thunk that one: 126,654, or 19*6666.

So, [a+b+c+d] = 19. And what digits of what we subtracted do we know? 126,654 - 122**0 = 00[3 or 4]**4. The [3 or 4] comes from either carrying the 1 from the hundred's place, or not.

So then, this simplifies to 3**4, because each of these 4 digits have to be different. 3+4 = 7, 3+4+*+*=19, so *+*=12. What single-digit numbers add up to 19, and make you carry the hundred's place in that particular subtraction? Well, maybe 7 and 5: let's try. 126,654 - 3754 = 122900. Yay! It worked. Are there any others? 4 and 8, but 4 has been used. 3 and 9, but 9 has been used. 2 and 10, but 10 is a two-digit number. 1 and 11, but that just don't work none.

So, the number subtracted was 3754. Hope that helps, it was fun
[/musings]

[lau_kai]

wow... amazing thanx again...

i had thought of breaking the sum up... but then i lost the plot after i did that...

did it take long to think of a method?

[jemidiah]

You're certainly welcome!

Not to sound like I'm tooting my own horn, but I pretty much just wrote the solution straight through. After thinking up the sum [which took about the same time as writing the first lines, for curiosity], the problem took care of itself.

It was interesting, though

[lau_kai]

wow... i see... so good... ill be good at maths one day... just another few years after i finish high school and tertiary studies...

how can you see a method straight away? for me... once i get stuck... a method never comes to me... but as for all you maths genius's... How the hell can you see it straight away?!?

makes me jealous... boo...

[jemidiah]

lol

Here's our secret: we forsake most every other area of life (that would be useful to us) for the thing that we're really good at, math.

[sorry for the chit-chatiness, but it's not entirely off topic]

[keithy]

Originally Posted by lau_kai

This is an utmost annoying question that i assume requires lots of patience(for people like me...)... or just a few secs for mathematicians...

Joe asked Penny to think of four different postive digits and add up the 24 different 4-digit numbers that could be made using the four digits. Then he asked her to subtract just one of the 4-digits numbers from the total and write down the new total. When asked what number she had written down, Penny replied: "122**0".

Joe didnt quite hear the 4th or 5th digits. But he was able to work out the number Penny had subtracted. What was that number?

well, is there any combinations of formluae that may work on this question? or can it be onli pure guess and check?

Thanx in advance

k....look dont ever do this again!!!!!!!!! this is striaght out of new scientist magazine uk.. it the enigma problem july 23rd issue...

with the solution underneath..... you suck if you hand this in with ur entry....if you couldnt do it yourself you should just keep trying till you do, you never learn how to do these problem by reading other peoples, you need the ability to think like that your self. get it by practice and perserverance.

if you came across this problem some other way i'm sorry (but i doubt it).
nxt time at least quote from where you got it from.

i worked it out that way, its not a hard problem, but now its on the web.. oh well.