## Cool Math Brain Teaser, from MichaelC

Was leafing through the stack of pending puzzles, and came across this cool math brain teaser I totally forgot to put up last year. It was created by math whiz MichaelC. Haven’t seen him around in awhile, hope he’s doing ok.

“I know the birth years of myself, my mother, and my sister are prime numbers. And I know when my mother celebrates her birthday this year, the sum of our ages will be 196 years. I also know my sister is 2 years older than I am.

But the question I have is, how old is my mother?”

Jimmy Anders| PUZZLE MASTER | Profile January 26th, 2010 - 12:42 amIf X is the birth year of the mother, Y is the birth year of the sister, then MichaelC’s birth year is Y + 2. Since the sum of ages is 196, and the puzzle is from 2009, this means:

196 = [2009 – X] + [2009 – Y] + [2009 – [Y + 2]],

which means:

X + 2Y = 5829.

Since X < Y (the mother was born before the sister):

5829 = X + 2Y 1943.

So, all we need to do is find “twin primes” whose smaller twin is greater than 1943, whose bigger twin is less than 2009, and which yield an X from the first equation that is prime. This is easiest checked by a list of primes, which can be found online, or generated by a homemade program using the sieve of Erastothenes, whichever you like. (You’ll find that there is only one other pair of twin primes in the range, but even if they did work, then the mother would’ve been impossibly older than her children.) The triple of primes in question turns out to be:

X = 1931, Y = 1949, and Y + 2 = 1951.

If the mother was born in 1931, that means her age in 2009 was 78.

ikickass| Profile January 26th, 2010 - 4:01 am1931

ikickass| Profile January 26th, 2010 - 4:02 amsorry,78 (or 79,we’re in 2010 now) is the answer

hex| PUZZLE MASTER | Profile January 26th, 2010 - 7:20 amRK, don’t tell me this lady cost you the same as the other girl (from ….olia)

hex| PUZZLE MASTER | Profile January 26th, 2010 - 7:48 amThe solution is possible if and only if either my sister or my birthday comes before mother’s.

mother: 1933

sister: 1949

me: 1951

so mother’s age: 77 after she celebrates her birthday this year, ie 76 before her birthday.

engjs1960| Profile January 26th, 2010 - 7:48 am1931, 1949, 1951, 78 years old

dhunnapotha| Profile January 26th, 2010 - 8:53 amMothers age is 78.

Assuming that “this year” in the question refers to 2009 (“I totally forgot to put up last year”, the birth years are 1931, 1949 and 1951

cheddarmelt| Profile January 26th, 2010 - 9:10 amThe only primes separated by two (self and sister) in the 1900’s are 1931/1933 and 1949/1951. Only the second pair results in a logical age gap. The sister was born in 1949 and the speaker was born in 1951, for a combined age of 120, if both have had birthdays this calendar year. That leaves 76 years, which would be 1934. Therefore, the mother was born in 1933; two of the family members have had birthdays already this year, but the third has not. Since the puzzle says, “when my mother celebrates her birthday this year,” indicating that she is the one who has not yet had her birthday. She is currently 76 and will be turning 77 later this year.

Tsopi| Profile January 26th, 2010 - 9:26 amseams impossible…i cannot find any prime numbers that have a distance of 2 numbers and when their sum is substracted by 196 get a prime number…unless i didn’t understand a part of the puzzle..this is impossible

Blusummers13| Profile January 26th, 2010 - 10:34 amJust a couple of questions:

Are the prime numbers the 4-digit years or the 2 digit?

Ex: 02 is a prime number, 2002 is not.

Also, what year is “this year”? I only ask since you mentioned that you forgot to post this last year.

Thanks.

Shawn| PUZZLE GRANDMASTER | Profile January 26th, 2010 - 10:39 amGood thing you said that this puzzle was from last year! The ages only work for 2009.

Mom = 78 years old (1931)

Brother = 58 years old (1949)

Sister = 60 years old (1951)

RK| Founder | Profile January 26th, 2010 - 11:13 amHi ther Blusummers13, the primes are the 4 digit years.

“This year”- is 2009

RK| Founder | Profile January 26th, 2010 - 11:27 amYes, please remember-this quesiton was posed in 2009, so you may want to check your answer as to ‘how old is my mother’

Blusummers13| Profile January 26th, 2010 - 11:48 amMother is 78. Born in 1931.

Blusummers13| Profile January 26th, 2010 - 11:50 amUnless she has had her birthday in 2010 already of course.

hex| PUZZLE MASTER | Profile January 26th, 2010 - 12:06 pmI noticed the “last year”, but thought you had adjusted the figures to compensate. Since it is 2009 now the answer will be slightly different:

The solution is possible if and only if my sister and my birthday come before or the same day as mother’s.

mother: 1933

sister: 1949

me: 1951

so mother’s age: 76 after she celebrates her birthday this year (2009), ie 75 before her birthday.

Glad that mother has become 1 year younger

Adam| Profile January 26th, 2010 - 12:40 pmI was born in 1951, and I am 58 years old.

My sister was born in 1949, and she is 60 years old.

My mother was born in 1931, and she will be 78 this year.

My mother is currently 77 years old.

suineg| PUZZLE MASTER | Profile January 26th, 2010 - 12:47 pmCool puzzle MichaelC, I think it goes like this:

two primes with a difference of two and that the ages of all the persons would fit logically with the sum of the ages being 196 for the year 2009:

Birth year

Mother: 1931

Sister: 1949

MichaelC: 1951

Ages in 2009:

Mother: 78

Sister: 60

MichaelC: 58

Verification: 78+60+58= 196

This would fit logically and all the birth years are prime.

fuzzy| Profile January 26th, 2010 - 2:22 pmMom must be 78. The kids are 60 and 58. Their years of birth are 1931 (mom), 1949 (sister), 1951 (“I”.

cheddarmelt| Profile January 28th, 2010 - 9:27 amI had assumed that RK would have recalculated everything to make it correct as of the day he posted. It never occurred to me that he would have done otherwise. That DOES change the answer.

Here are my new calculations:

The only primes separated by two (self and sister) in the 1900’s are 1931/1933 and 1949/1951. Only the second pair results in a logical age gap. The sister was born in 1949 and the speaker was born in 1951, for a combined age of 118, if both have had birthdays this calendar year. That leaves 78 years once she reaches the birthday in 2009, or 77 years of age at the time.

RK| Founder | Profile January 28th, 2010 - 9:48 amWell, I really botched this one, sorry about that.

MichaelC submitted this in 2009, and in my rush to put it up, I didn’t modify the #s to make it appropriate for 2010. When I realized the mistake, tried to point things out by adding a ‘comment’, but don’t think many of you saw it in time.

In any case, MichaelC was out to trip you guys up with a fiendish little maneuver

As it would have been presented in 2009,the correct answer would have been 77, not 78. (“When my mother celebrates her Birthday this year).

I think Chedddarmelt and Adam picked up on this

RK| Founder | Profile January 28th, 2010 - 9:55 am@Hex-:lol:

unlike wine, some things do NOT get better (more $$) with age!

hex| PUZZLE MASTER | Profile January 28th, 2010 - 11:10 amI have no clue how 1933 crept in my answer for the mother’s birth year, especially that my calculation paper shows 1931. Mother has become 2 years older than I had thought. Damn it!

Anyway, a prerequisite of the answer is that my sister’s birthday and mine must be earlier in the year than my mother’s. Otherwise, the puzzle will not have a solution.

michaelc| Profile January 28th, 2010 - 8:07 pmGreetings Everyone,

I’m a few puzzles behind as of late, but I swear I will catch up one weekend soon! There are a few I can’t wait to try.

The 77 part is a rotten trick. Almost as rotten as a 2009 birthday puzzle in the year 2010! Just teasing RK. Thanks for posting!

Keep up the great work with the site!

RK| Founder | Profile January 29th, 2010 - 10:07 amHey there Michaelc!! Thanks for stopping by!

10dulkar| Profile March 3rd, 2010 - 1:13 pmHi guys & gals…

I wonder why you all forget that 1949 is NOT a prime year! Why? Off course I’ll elaborate. 49 is divisible not only by 1 & itself but also by 7 so NOT a prime number…so all those answers involving births in 1949 are WRONG! Take that from me…

Now my solution is as follows:

Mother’s birth year: 1911

Sister’s birth year: 1959

My birth year: 1951

First and foremost I am assuming that the ‘current year’ (in which the puzzle is written) is 2009.

Now let’s start by listing the prime numbers (between 1 and 99). They are:

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71

73 79 83 89 97

Immediately we see that the only possible combinations for the sister’s & my birth years (in that order) are:

a. 3, 5

b. 11, 13

c. 17, 19

d. 41, 43

e. 59, 61

f. 71, 73

We can rule out the first 4 combinations (a through c) because the brother & sister would be so old that the mother would have to be younger (physically impossible!)

That leaves e & f. Let’s go through these 2 options and rule out the incorrect ones.

Take f.

Sister born in 1971 > is 2009-1971 = 38 yrs old

I am born in 1973 > am 2009-1973 = 36 yrs old

That would mean Mother is 196-38-36 = 122 yrs old

If mother’s 122 years old her birth year is 2009-122 = 1887

So now we have the following facts

Mother born in 1887 (11 is NOT prime)and is 122 years old

Sister born in 1971 (71 is prime) and is 38 years old

I am born in 1973 (73 is prime) and am 36 years old

The total age of the 3 of us is: 122+38+36 = 196, BUT THE MOST IMPORTANT POINT ABOUT THIS SOLUTION IS THAT THE MOTHER’S BIRTH YEAR (87 as in 1887) IS NOT A PRIME NUMBER…SO THIS IS NOT THE RIGHT SOLUTION

Take e.

Sister born in 1959 > is 2009-1959 = 50 yrs old

I am born in 1943 > am 2009-1961 = 48 yrs old

That would mean Mother is 196-50-48 = 98 yrs old

If mother’s 98 years old her birth year is 2009-98 = 1911

So now we have the following facts

Mother born in 1911 (11 is prime)and is 98 years old

Sister born in 1959 (59 is prime) and is 50 years old

I am born in 1961 (61 is prime) and am 48 years old

The total age of the 3 of us is: 98+50+48 = 196

The (ONLY) CORRECT OPTION leading to the answer is combination e

Regards,

10dulkar

10dulkar| Profile March 3rd, 2010 - 1:20 pm1. Oh please strike out the comment in my post above that reads:

My birth year: 1951

The correct line should read:

My birth year: 1961

2. Also in the section that reads:

Take e.

Sister born in 1959 > is 2009-1959 = 50 yrs old

I am born in 1943 > am 2009-1961 = 48 yrs old

That would mean Mother is 196-50-48 = 98 yrs old

If mother’s 98 years old her birth year is 2009-98 = 1911

my birth is not 1943 but 1961…

Sorry for the confusion…and this is the correct answer!

10dulkar

Hex| PUZZLE MASTER | Profile March 4th, 2010 - 10:34 am10dulkar, why do you consider only the last 2 digits of the year when checking for primeness?

10dulkar| Profile March 4th, 2010 - 5:06 pmHex…

That is a good point. Guess one can look at it both ways, but the 4 digit way is the true way. That results in the following solution:

Sister > born in 1949 > is 2009-1949 = 60 years

My age > born in 1951 > is 2009-1951 = 58 years

Mother’s age = 196-60-58 = 78 years

Mother’s birth year = 2009-78 = 1931