A Very Good Year
Math brain teasers require computations to solve.
The year 1978 has an unusual property. When you add the 19 to the 78, the total is the same as the middle two digits (97). What will be the next year to have this same property?
Answer
The next year to have the same property will be 2307.
23 + 07 = 30
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Comments
air_up_there_jr
Sep 22, 2007
 nice.. how did you think of this? 
MellowWillow
Sep 23, 2007
 Interesting. 
jasmin215
Oct 28, 2007
 boy!, you had me here with the calculator trying to add numbers...gave up at 1985
cute 
(user deleted)
Dec 11, 2007
 What about '2042' 
(user deleted)
Dec 11, 2007
 Never mind the last comment, sorry 
avonma
Dec 21, 2007
 Sorry. I'm not very good at numbers... 
surintan
Dec 21, 2007
 After year 2307
Subsequently you can just increase the second and third digit by 1 each for the next 6 numbers, i.e.
2417
2527
2637
2747
2857
2967 
Nomez
Dec 21, 2007
 a*1000+b*100+c*10+d = the year
b*9 = a*10 + c*9 + d
Any a,b,c and d that are greater than zero and satisfy this equation will fit this puzzle
Of course, this alone won't satisfy the puzzle. Finding the lowest one takes a little bit of trial and error.
I actually got 2417 the first time I tried it. That's because I forgot to account for 0's
Cool puzzle. 
gghali
Dec 21, 2007
 It's actually a bit more defining than that:
You know that the number can be represented as abcd. We also know that ab+cd = bc
So, there are 2 possibilites:
b+d
b+d = c
a+c = b
b+d >= 10 (we need to "carry the 1")
b+d = 10+c
a+c+1 = b
In the former case, combining the formulas leads to b+d = ba, which cannot work unless a=d=0, which doesn't apply since a>0 (197. So we know that it is the latter case.
Again, combining formulas, we get
a+c+1 = 10+cd
a+d = 9
so we know that years that work must be of the form
1xx8
2xx7
3xx6
...
Plugging this back into b=a+c+1, we see that in the case of a=1, the second digit must be 2 greater than the third digit. Clearly, the last of the years 2000.
So we know it is of the form
2xx7, and that b=c+3. So the minimum values for b and c are set when c=0 > b=3.
Thus, 2307. 
coolguy5678
Dec 21, 2007
 I did exactly the same thing as Nomez 
leftclick
Dec 21, 2007
 int i;
for (i=1979; ((int)Math.floor((double)i / 100.0) + (i % 100)) != (int)Math.floor((double)(i % 1000) / 10.0); i++) { }
System.out.println(String.format("Found a match: %d", i));

leftclick
Dec 21, 2007
 Oh yeah, the output...
Found a match: 2307 
bradon182001
Dec 21, 2007
 Yeah right. I agree with all of the above since I have no clue. This one must have taken a great deal of thought. Good job Soccerguy. 
dstarbird
Dec 21, 2007
 Darn...I got 2417, which is the one after that. Owell... 
leftclick
Dec 21, 2007
 Oh yeah and a slightly modified program gives the full list: 1208, 1318, 1428, 1538, 1648, 1758, 1868, 1978, 2307, 2417, 2527, 2637, 2747, 2857, 2967, 3406, 3516, 3626, 3736, 3846, 3956, 4505, 4615, 4725, 4835, 4945, 5604, 5714, 5824, 5934, 6703, 6813, 6923, 7802, 7912, 8901
1978 was a great year, btw Full marks to this teaser 
tommysmo
Dec 21, 2007
 i didn't feel like doing the math this morning. 
UlsterCharlotte
Dec 21, 2007
 Meh, I got 2747... I'm happy enough just to get a number that was right. Very cool puzzle, though. 
phyllisa
Dec 21, 2007
 Intersting but I agree with tommysmo. 
FatHead
Dec 21, 2007
 Wow, surintan and Nomez, I just used Guess and Check. 
lfishbach
Dec 21, 2007
 This is a classic Diaphantine? Problem. Let us find all the years starting with a 2 that fit the solution. The year can be written as 2ABC. Then 20+A +10*B +C must equal 10*A + B. Thus
9*A = 20 + 9*B + C. A, B and C MUST BE integers. Dividing by 9
A = 2 + B + (2+C)/9 and C 
mumbojumbo1234
Dec 21, 2007
 2031 ;x 
mumbojumbo1234
Dec 21, 2007
 Oh heh misread it as first and last together get middle. 
mi2mo2tx
Dec 21, 2007
 ouch  too tough for my brain. kudos to all who attempted and then got it!! 
UptheHill
Dec 21, 2007
 That was too hard, even if coffee was available!!! 
dogg6pound9
Dec 21, 2007
 wow... i need to go back to school after reading nomez and suritans algebraic solutions... i came up with 2857 based on the "guess and check" theory, but obviously it doesnt come next... i couldnt find a logical pattern to solve it, or think of an equation to help... i highly commend those who know their algebra to solve this quickly and accurately 
dogg6pound9
Dec 21, 2007
 i forgot to comment about the teaser ... this is the best teaser i've seen so far... 5 *'s to soccerguy for thinking up this one... and i have to ask on behalf of myself and i'm sure everyone else... how did you come up with this??? 
nerdyiscool
Dec 21, 2007
 Didn't like it... but I'm a much more linguistically minded individual. Math frightens me, and I personally have several friends two years younger than me who could sooo kick my butt in maths. I'm sure it was great, just not my thing. 
auntiesis
Dec 22, 2007
 ? Huh ? 
mondayschild59
Dec 22, 2007
 I didn't even try. Math is not my thing. Trivia is my forte. Thanks for posting. I know the others enjoyed it.
Monday~ 
javaguru
Jan 08, 2009
 I was certain there was a solution that started with 2, so
2A + CD = BC
[(B + D) / 10] + 2 + C = B
and
(B + D) % 10 = C
(where [x] is floor xthe integer portion of xand % is the mod, or remainder, operator)
[(B + D) / 10] is either 0 or 1.
This gives:
(B + D) % 10 = B  [(B + D) / 10]  2
If B + D is less than 10, then B + D = B  2, which is clearly not possible, so B + D >= 10. So now
B + D  10 = B  3
D = 7
Also B >= 3. Plugging 3 in for B gives
C = (3 + 7)  10 = 0
2307

javaguru
Jan 08, 2009
 The first equation in my previous post should be
2B + CD = BC 
javaguru
Jan 08, 2009
 By the way, why are so many people that don't like/aren't good at math commenting on a math teaser? It's not like not being good at math is something to wear with pride. 
opqpop
Sep 11, 2010
 This wasn't really hard.
We can quickly see 1979, 198X, 199X isn't going to work, so we go into the 2000's.
The first 2 digit number is already 20, so we skip to 22XY (X and Y not necessarily distinct) with X > 2 because we need the middle two digit number to be > 22. Obviously, this isn't going to work for any X, so we skip to 23XY.
Starting with X=0, we immediately see 2307 works and we're done. 
opqpop
Sep 11, 2010
 After reading the comments, I can see that my "bruteforce" solution turned out to be very quick due to some luck...
hurray 
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