Digits and Years

Number Theory Level 2

The year 2011 has the property that the year plus the sum of its digits is equal to 2015, since 2011 + ( 2 + 0 + 1 + 1 ) = 2015. 2011 + (2+0+1+1) = 2015. Find the only other year with this property.


The answer is 1993.

Magic Math by Magic Math , 21, Thailand

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7 solutions

Eli Ross Staff
Sep 16, 2015

It is easy to verify that none of the years from 2000 onward will work, so let's check in the 1900s. There are many ways to go about this search that will work. Here's one possible approach:

If the year is 19 A B 19AB where A A and B B are digits, then we need 1900 + 10 A + B + ( 1 + 9 + A + B ) = 2015. 1900 + 10A + B + (1+9+A+B) = 2015. This simplifies to 11 A + 2 B = 105. 11A + 2B = 105. Since 2 B ≤ 2 ⋅ 9 = 18 , 2B \le 2\cdot 9 = 18, we must have A = 9 , A = 9, which gives B = 3. B=3.

Thus, 1993 is the other year that satisfies this property, with 1993 + ( 1 + 9 + 9 + 3 ) = 2015. 1993 + (1+9+9+3) = 2015.

Remark: It is easy to show that no earlier century could work since the largest sum of digits for a four digit number is 4 ⋅ 9 = 36 , 4\cdot 9=36, so the year must be at least 2015 − 36 = 1979. 2015 - 36 = 1979.

8 Helpful 0 Interesting 0 Brilliant 0 Confused

Brilliant, thanks for the solution

Josemaria Loperena - 5 years, 9 months ago
Sadasiva Panicker
Sep 17, 2015

1993+(1+9+9+3) = 2015

3 Helpful 0 Interesting 0 Brilliant 0 Confused
Ahsan Azhar
Sep 17, 2015

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Jim Gao
Sep 17, 2015

0 Helpful 0 Interesting 0 Brilliant 0 Confused 

1
2
3
4
5
 
int X, R;
for (X=0;X<2015;X++){
   R = X + ((X/1000)%10) + ((X/100)%10) + ((X/10)%10) +(X%10);
   if (R==2015) printf("\n %d",&X);
}

Junio Cesar Ferreira - 5 years, 9 months ago

It is easy to verify that none of the years from 2000 onward will work and when you try 1999 it will be 1999+28 =2027

that mean its 12 more than 2015 so we will need to sum it from 2027

but if you decreased any from that year or the sum of its digits the other will be decreased by one in the 9 times after 1999

so you sum (12/2) from each

so 1999-6 =1993

1+9+9+9-6 =22

1993+22 =2015

sorry for my english and because its not a general solution

0 Helpful 0 Interesting 0 Brilliant 0 Confused

excellent solution

Arun Joshi - 5 years, 9 months ago
Chris White
Sep 17, 2015

I don't get questions like this that are easier/faster to do by hand than by a formula. Just count every year less than 2011 until you get the right answer 1993... takes 2 minutes tops.

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Renato Silva
Sep 17, 2015

10 for a=0 to 2

20 for b = 0 to 9

30 for c = 0 to 9

40 for d = 0 to 9

50 let x = 1001 a+101 b+11 c + 2 d

60 if x = 2015 then print a;b;c;d

70 next d

80 next c

90 next b

99 next a

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Pseudocode

Rameel Shafqat - 5 years, 9 months ago

Meanwhile, 5 minutes ago....everyone else got it already.

Kerry Lasiter Billingsley - 5 years, 9 months ago

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