Grandfather's Age: Newsletter Challenge (02/27/07)

Assume all ages are positive whole numbers.

(1) G = J(J+1)/2 - 1

(2) G = Tx10 + U {T = Tens; U = Units}

(3) J = T + U

(4) G = (10,11,12...99) {Double Digits only}

To satisfy (1) & (4) 4 < J < 14.

Only J= 5, 8, 11 satisfy (2) & (3).

Possible solutions are :

J = 5, G = 14

J = 8, G = 35

J = 11, G = 65.

Given that Joe was able to perform basic maths functions I assume he is a human. To allow time for two generations between G & J I'd say its most likely that Joe is 11. Grandfather is 65.

Re (4): Now really, does the problem actually state that the two digits of the grandfather's age are the same digits?

-e

Guest #1 is correct! READ NO FURTHER!

...actually check out STL Engineer Post #10.

'Tis true, well said that man!

Now, more challenge questions please...

Well, not exactly. Guest #11 has another solution, improbable, but not impossible!

i concur with guest......11 and 65...

I agree with all equations except (1). If I am reading the question correctly, it states that Joe's grandfather's age is 1 less than the number of birthday's Joe has had plus his age - not the sum of the ages of each birthday. Therefore, if Joe was 11 then he has had 11 birthdays and Joe's grandfather's age would be 21 (11 birthdays + 11 years old - 1). (Obviously not the correct answer but given to restate equation (1) to read, G = 2J-1.) Someone please confirm I am reading this question correctly before solving. Thanks.

Ok, this situation sounds like a case of 3-equations-and-4-unknowns. So, if you added up the son's possible ages, you would get: {sum(aj), j=1 to n}=

0+1=1; +2=3; +3=6; +4=10; +5=15; +6=21; +7=28; +8=36; +9=45; +10=55; etc.

The possible ages of the grandfather would then be one year less:

sum(aj)=Ag+1 => Ag=sum(aj)-1= 0; 2; 5; 9; 14; 20; 27; 35; 44; 54; 65; 77; etc.

And the grandfather's age must also be: Ag = 10 x A10 + 1 x A1 = 10A10 + A1

Further reduction would show; sum(aj) = 1 + 9 x A10 & A10's must be integers, so

possible sum(aj)'s would then be, for: (1)=>17; (2)=>27; (3)=>35; (4)=>44; etc.

which has a (first) match of the value for sum(aj) = 65, as found previously.

(subsequent age matches would be unrealistic, at the present time...)

So, checking for "65"; 6+5=11 <=?=> (1+2+3+4+5+6+7+8+9+10+11)-1=65.

Which also shows the son is 11 years old and the grandfather is 65 years of age.

(admittedly, this way is not very mathmaticaly "elegant", but it is "serviceable"...)

I guess 14 and 95

If the kid's age was 14 then grandpa would need to be 104 (unless I missed something).

You missed something:

"Joe also discovered that if he added the two digits of his grandfather's age, the result was his own age."

The grandfather cannot be 104, because that is three digits, not two. "10" is not a digit.

Thanks for the insite.

When I posted #5 in response to #2 it was just to point out that 95 is the wrong age for Grandpa when the kid is 14.

I really don't think Gramps is 104. It's just what the math says if the youth was 14. that's all.

We-e-e-l-l-l-l-l-l,.....you could have said that although, if Grandpa was 95, the kid might be 14 because 9+5=14, he could not be 14 because 1+2+3+4+5+6+7+8+9+10+11+12+13=14=105, not 96.

There are four variables and three equations, so there doesn't appear to be a unique solution. Among the three integer solutions satisfying the given criteria I'd have to say that Joe is 11 and Grandpa is 65, human nature being what it is.

On the other hand, Joe and Grandpa could be some effing smart horses ages 8 and 35. Mr. Ed knew how to use an HP-65. Did you know that?

Or a couple of canine prodigies, ages 5 and 14? Naturally, this rules out very-pretty-but-dumb-as-a-box-of-hammers Afghans. And Shitzus. Cannon-fodder.

Cats on steroids?

Nah. They're too smart to bother.

-e

Joe age should be 10, Grandpas should be 55 because 1+2+3+4+5+6+7+8+9+10=56 which is 1 year more than grandpa's age and also Two digit addition of Grandpa age is (5+5)=10 which is the current age of Joe.

Unfortunately, adding 1+2+3+4+5+6+7+8+9+10 = 55, not 56.

I apologize for the terrible mess in totalling

A agree with guest No. 1. Joe's age has to be 11 and that of gradpas 65. I think this time my addition is correct. 1+2+3+4+5+6+7+8+9+10+11=66 which is 1 year more than grandpa's age and also two digit addition of Grandpa age is (6+5)=11 which is the current age of Joe.

11 and 65.

Congratulations! You came up with the same answer as Guest #1!

At least you could congratulate him and acknowledge that he had it first, otherwise your post is totally irrelevant, as you came up with nothing new!

How many more times will we treated to the same answer? If you have no comments or nothing new to add, why post the same answer?

Try a little bran with your breakfast.

I guess 11 and 65.

Joe is 8 and his Grandfather is 35

1,2,3,4,5,6,7,and 8 added together is 36 ( 1 more than his Grandfathers age)

3+5 is 8

8, 53

"1,2,3,4,5,6,7,and 8 added together is 36" as noted above, NOT 54 (53+1)!

Does Joe play the banjo? If he is 8, then Grandpa must have been about 14 when he impregnated Joe's Grandma, and Joe's Mom or Dad, their child, must have been about 14 when Joe was born.

If this is so, chances are Joe lives in the inner city of a major metropolis, or in Arkansas.

I think your post is uncalled for commentary, and should not be allowed on this forum. The question makes no assumptions of inner-city people or of Arkansas, of which place I am not from, but would be insulted if if I did. It is a typical middle school problem found on many test to try to get people to exercise the brain a little.

Gyaackgk!! I've been arrested by the PC Police!

I think that the post is called for and should be allowed on this forum. You would be insulted if you came from there since it is an insult. It may be a middleschool problem but, the answer is a feasible answer where it does not teach the children that having kids at the age of 8 to 13 is common. If you want to promote teenage mothers in inner city people and people of Arkansas, go ahead.

Oh, fie!

You cannot possibly be serious with this answer. Although it mathematically meets the criteria ([∑n]-1 = 10T +U; T+U = n), it proposes that we believe in two generations of impossibly precocious fathers. Their average age must be less than twelve and a half. Remember that it takes 40 weeks to come to full term.

Please, just believe the solution of the initial responder. An eleven year old boy with a 65 year old grandfather is both mathematically correct and physiologically believable.

If (8,35) is an answer, at least one of J's dad and grandpap must have their child at or before their age of 13.5, right? Should the reality be considered in the solution or not? Or just a math?

Joe: 9

Grandpa:45

"Missed it by that much!" , Maxwell Smart, Agent 86.

The sum of Joe's birthdays, if he is 9, is 45, but the challenge question says it must be one more than his grandfather's age.

Close, but NO CIGAR!

10 and 55 or 56 does not work either.

Joe is 11 (1+2+3+4+5+6+7+8+9+10+11=66), Grandfather is 65 (6+5=11)

You are both correct and irrelevant. Please read posts #1, 8, and 10, and then report to the Department of Redundancy Department.

Thank you.

NEXT!!!

Must.......resist......temptation.....to.......view.....further......posts.....on.......this.......thread!!!!!!!!!!

64 and 10

Half right. Total of birthdays (if age 10) is 55, not 65.

10 b-days

+ 10 yrs old

- 1

64 and 10

Joe is 10 and his grandfather is 55.

If Joe is 10, sum of his birthdays is 55, but then grandpa should be 54, not 55.

A trick may be that Joe adds all birthdays INCLUDING his age now. That means the total is N/2(N+1) + N. If Joe is 10, that gives 5(11) + 10 = 65 which is his grandfather's age plus 1 so the grandfather is 64. Adding the grandfathers two digits = 10 which is Joe's age.

If there was no trick, then the total of Joe's birthdays is N/2(N+1) = 66 (If Joe is 11) so the grandfather would be 65 (and 6+5=11)

There is no trick, and the other answer has been given many times.

Thank you for playing, "What's my redundancy?"

Please enjoy our home version.

Now Mr. STL, don't get your panties in a wad. Sit back and enjoy the show. Here, have a beer.

Now many, many years ago, when I was twenty-three,
I was married to a widow who was pretty as could be.
This widow had a grown-up daughter who had hair of red.
My father fell in love with her, and soon they, too, were wed.

This made my dad my son-in-law and changed my very life,
My daughter was my mother, cause she was my father's wife.
To complicate the matter, even though it brought me joy,
I soon became the father of a bouncing baby boy.

My little baby then became a brother-in-law to Dad,
And so became my uncle, though it made me very sad.
For if he was my uncle, then that also made him brother
Of the widow's grown-up daughter, who, of course, was my stepmother.

Father's wife then had a son who kept him on the run,
And he became my grandchild, for he was my daughter's son.
My wife is now my mother's mother, and it makes me blue,
Because, although she is my wife, she's my grandmother, too.

Now if my wife is my grandmother, then I'm her grandchild,
And everytime I think of it, it nearly drives me wild,
For now I have become the strangest case you ever saw
As husband of my grandmother, I am my own grandpa!

What a facinating tale of troubles you got involved in getting married to a window, and in the process became the grandpapa. By the way what is your age?

You mean 'widow?' I'm a diehard Linux & Mac user/programmer, having sworn off WinDoze some time ago (and not soon enough). As for my age, ask my prodigal grandson.

-e

the age of Joe must be included: "he found that if he added up all the birthdays he has had, including his age now, the result was one year more than his grandfather's age"

Yes, the age of Joe must be included, but only ONE time, his current age ON his last birthday. Please note he added up "all the birthdays", which included his age now, not "all the birthdays plus his age now", which would be different.

Example, if Joe was three, he would add 1+2+3=6, not 1+2+3+3=9.

A rookie mistake.

Well that is certainly one dead horse, anyone else care to beat it? Good job 1.

What ages are Joe and his grandfather ? JOE grandfather 1 year 2 years 3 years 4 years 5 years 6 years 7 years 8 years 9 years SumBirthdays 10 years 55 11 Age now Sum Birthdays + Age Joe Age 60 66 65 6,5 6 5 JOE grandfather 11

That makes no sense. Were the numbers in columns and you lost your formatting?

Joe's age is 9 and his granfather's is 54

1 + 2 + 3 + ... + n-1 + n = n(n+1)/2 (note 1). Substituting your estimate of Joe's age for n, we have 9(9+1)/2 = 45. Now the problem stated that the sum of Joe's birthdays, including Joe's present age, equals Grandpa's age plus 1. Granda's age must therefore be 44 - not 54 - according to your estimate of Joe's age. But then Joe would have to be 8 years old, as the challenge also states that the sum of the two digits of Grandpa's age equals Joe's age, or 4 + 4 = 8. But you asserted that Joe is 9, a contradiction. Take a closer look at Guest #1's post to see how it works.

==============

Note 1:

By the way, for any CR4 members and guests who might be wondering how Guest #1 knew that

1 + 2 + 3 + ... + n-1 + n = n(n+1)/2, here's how...

Write down the series from one to 'n' (in the case above, Antorone asserts that Joe's age is 9, so, according to the challenge we'd sum the first nine integers. This is not a problem with little numbers such as we have here, but what if we wanted to add the first 10e187 integers? Big problem, so we need a shortcut like Guest #1 used).

1 + 2 + 3 + ... + n-2 + n-1 + n = x. Now underneath this series write down the same series, but with the terms in reverse order, like this:

1 + 2 + 3 + ... + n-2 + n-1 + n = x
n + n-1 + n-2 + ... + 3 + 2 + 1 = x.

Now, if we add the two (numerically identical) series together, term by term, we get...

n+1 + n+1 + n+1 + ... + n+1 + n+1 + n+1 = 2x

But wait! All we have now is just n each "n+1" identical terms? Sweet! Now we can really make short work of it: n(n+1) = 2x.

Cool, huh? But we're not quite done.

Since we added two identical series together, n(n+1) must be twice the desired sum, so, dividing both sides by two, we get what we came for: x = n(n+1)/2.

WTG, Guest!

-e

Joe - 11 years

Grandfather - 65 years

We are talking about Joe´s birthdays and not about the anniversaries, therefore we must count like this:

1-st birthday means 0 years of Joe´s age and so on.

Due to this presumption the possible results are:

G=44. J=8 and G=54, J=9.

If the grandfather got his son (Joe´s father) when he was over twenty and if Joe was born when his father was over twenty then the result should be:

G=54, J=9.

Bye Martian

1-st birthday means 0 years of Joe´s age and so on.

-----

Interesting. I've heard that the Chinese (at least some of them) include time spent in the womb when calculating someone's age. This makes sense, of course, as the difference is only one of location. Perhaps whether birthdays - considered, in the U.S. at least, as being synonymous with anniversaries - should be distinguished from anniversaries is something to ask the Framer. In other parts of the world birthdays are considered distinct from anniversaries. Here in the States one's first birthday is the first-year anniversary of one's birth. Strictly speaking, I suppose, one could consider the day on which one is born as one's first birthday (which it is, after all, as none have preceded it), with one's second birthday marking the beginning, rather than the end, of one's second year post partum. Perhaps the framers of the Challenge question could be persuaded to clarify? Like, how old is a child on her "first" birthday? Zero years or one year?

-e

You are waaaaay overthinking this man. Kind of like the people who said the new millenium started in 2001, not 2000, since the first millenium began in year 1. All I know is that when my boy was 12 months old, everyone sent him birthday cards that said "first birthday", and when he turned 11 they said, "11th birthday". So your thinking, though logical, is not practical, and should NOT be used in this context.

Joe's age is 11 years and his grandfather's age is 66 years

Le Grand-père a 65 ans, et Joe 11 ans

sum of the first n values = (n+1)*n/2

1 1 1

2 3 3

3 6 6

4 10 1

5 15 6

6 21 3

7 28 10

8 36 9

9 45 9

10 55 10 true

11 66 12

12 78 15

13 91 10

14 105 15 105more than 2 digits

the answer wil be 10 yeasr old for Joe and 55 years old for his grandfather

The answer cannot be 10 and 55 since the addition of all birthdays plus current age of Joe should add up to a figure which is one year more than Grandpa's age. That is why the answer is 11 and 65.

joe is 11 and his grandfather is 65 add 1,2,3,4,5,6,7,8,9,10 and 11=66 66-1=65 add 6+5=11.

Yes, Joe is 11 and his grandfather is 65. I got it by trial and error. If Joe had been older so the numbers were larger this method would take too long.

Yes. Like, if Joe were around 100,000 years old, computing his and Granpa's ages would take nearly two full nanoseconds on a fully-configured IBM Blue Gene. Running under Linux, of course. (Under MicroLimp Vista we'd have to call it the IBM Blue Screen)

-e

Joe is 11 and his grandfather is 65. Joe's ages added up for 11 years total 66 which is one more than his grandfather's age.

Lets remember that smart allic little kid named Gauss who came up with the answer to 1+2+3+...100. take the first and last numbers and add them 1+100=101, then the second one on either side, 2+99=101... and the third ect. they all equal 101. There are 50 sets of 101 or 50x101=5050. If we apply that to the kid's age starting with about 10, and play with it a little we get 1+10=11 11x5=55. Almost there. Lets keep adding a year to the kid. 55+11 years old makes 66. 66-1= 65 Grandpa's age!

Yes. Please see Post #41.

Gauss' instructor told him, as punishment, to add the first 100 integers. No problemo. At age six, Gauss did his father's payroll.

How do people keep getting this answer wrong??? As stated above, there are no tricks to it and the first guy got it right. But still, people come up with 9 and 10 which are incorrect answers!

Boneheads! Must be those inner city or Arkansas folk. Haha!

8 and 35.

Kelly

Just to add a slightly different approach to this, I prefer to try and solve these puzzles in a programmatic way, although the answers the same as a number of other people have found.

In Excel add in Cell A1 'Boys Age', Add in Cell B1 'Grandfathers Age'

under the Cell 'Boys Age' in the first column add the numbers 1 down to 20.

In column B under the Cell 'Grandfathers Age' add :

=IF(SUM(A$2:A2)-A2>32,IF(SUM(A$2:A2)<100,IF(MOD(SUM(A$2:A2)-1,10)+INT((SUM(A$2:A2)-1)/10)=A2,SUM(A$2:A2)-1,""),"" ),"")

Copy this down all 20 rows.

The only cell with writing in column B will be the number 65 against the Boys age of 11

It presumes that the grandfather, and his son, were both at least 16 when they had children, and also that the grandfather must be less than 100 as his age can only have two digits.

I tried to follow your formula but was unsuccessful.

very clever, i tried it, it worked.........

While y'all are wastin' time with fancy math, I did some cypherin'. Granpa's maximum age is 99 (two digits) and 9+9 = 18, so Joe can't be older than 18 years old. So how hard can it be to add 1+2+3+4... for every year up to 18? When I do that, I find that Joe can't be older than 13, since the sum of birthdays to 14 years = 105, which is > (99+1 = 100). Now it's even easier: 13 years old doesn't fit, 12 years doesn't fit, but hallelujah !, 11 years does. The sum of years from ages 1 to 11 = 66 and 66-1 = 65 and 6+5 =11. All that in less than 2 minutes without a book or computer.

I did basically the same thing, but I did use a calculator for cipherin' all them big numbers when adding the birth days.

Also 11 and 66

Kelly

The Turbine Doctor says that Joe is 11 and his grandfather is 65.

Joe was thinking about his grandfather's age; he found that if he added up all the birthdays he has had, including his age now, the result was one year more than his grandfather's age. Joe also discovered that if he added the two digits of his grandfather's age, the result was his own age. What ages are Joe and his grandfather?

Definetely, Joe's age is 11. And his grandparent's is 65

1+2+3+4+5+6+7+8+9+10+11= 66

so, his grandfather age is: 66-1= 65

65 = 6+5 = 11

Both ages are reasonable and might be true.

The problem does not state that Grandpa is still alive, so has anyone figured that Joe could be as old/older than his Grandpa?

Joe is eleven, his grandfather is 65.

Joe is 11; Grandfather is 65 years old.

11 factorial equals 66, 66-1=65 and 6+5=11!

Finally in post #72 someone referenced the word "factorial". Why it took 72 posts, I don't know. Takes about 30 seconds to do this problem in your head if you realized that the sum of ages was describing Joe's age factorial.

To all who are still posting incorrect answers - STOP!

While I agree with your sentiment: To all who are still posting incorrect answers - STOP! , I cannot agree with your use (or the use by #72) of the word "factorial" to describe the summation of Joe's ordinal birthday numbers. A "factorial" is the product (multiplication), not summation (addition). From the Wikipedia:

"In mathematics, the factorial of a positive integer n is the product of all positive integers less than or equal to n. "

The additive analogue to "factorial" is the "triangular" number. Hence, the "triangular" number of n=11 is 1+2+3+4+5+6+7+8+10+11=66

For more info on "triangular" numbers, here is the Wikipedia link:

http://en.wikipedia.org/wiki/Triangular_number

To be mathematically correct there is 4 solution:

Joe Gr.d

2 2

5 14

8 35

11 65

If we take that both of them live in society where genetic engineering is on its peak, all solutions are possible. ( Gr.d is just recently genetically duplicated ) Also ,if grandfather is recently reincarnated , we must consider all solutions. :)

This is an impossible question because a person has only one birthday! The rest are anniversaries.

Of course it is and they are! But a Coke is a "tonic" in the northeastern United States, "pop" in the Midwest, a "soda" in the south-central U.S., a "coke" in other regions - even if it's a Pepsi - and a post-partum anniversary is a "birthday" from coast-to-coast. Say "anniversary" here in the U.S. and people will wonder if your marriage in the maternity ward was a shotgun wedding.

-Professor Henry Higgins

we assume the age of boy =X

THEN grandfather =J

J=(X0+X1+X3+.....+Xn)-1

where n:birth day for boy now

Just to be awkward, I tried reading the question as near literally as I could, and adding the birthdays instead of the age at the birthdays. If I count the birth as a birthday, I get 2J+1=10T+U+1, and J=T+U. This gives U=8T, so grandpa is 18 and Joey is 9. So cloning (or sperm extraction followed by accelerated ripening and artificial insemination) seems to be involved. Highly irregular...

I tend to somewhat agree. If you are 6 months old (0 years) you have had one birthday. Therefore when you are one-year-old have you had two birthdays? I have always been told that everyone just has one birthday and each additional year is merely an anniversary of your birthday. (After all, peole don't celebrate their "75 weddings".) I dislike these TV shows that broadcast "First Birthdays" while showing one-year-olds.

Do not forget he is adding his birthdays and age also.

I guess 11 and 65.

This question and exchange made me think. I think that those who act the smart aleck and rudely attempt to intimidate thinking outside the box should keep on sending but pull the plug first.

I think this is a very poor question due to the many undefined variables as noted by several who were not intimidated or redundant.

I may have missed some but I applaud #43 Martian, "Most of us are born one time at zero years of age." #73 Marijana -Outside the box with four solutions, #74 Guest who pointed out that the colloquial term birthday is actually an anniversary of our only birthday. #80 Guest who felt he was being awkward by reading the question literally.

To come even close to answering this quiz we might agree that "birthday" is the day we celebrate another anniversary of our birth.

Nowhere can I find instructions to add the accumulated ages at consecutive birthdays (1+2+3 etc.). This may be a leap of logic. It merely says add up all the birthdays 1+1+1 etc.)

If Grandpa has two digits in his age he has to be 10 or over.

If Joe is 10 and has had 10 birthday anniversaries then 20 - 1 = 19 for grandpas age and 1+9 = 10 to satisfy the two digit requirement.

I also applaud those of you who suggested alternate reproductive methods instead of staying inside the box and citing impossibilities that weren't imposed and don't exist.

Let us encourage each other to think more abundantly and put criticism where the sun never shines.

Yes, and we could raise other questions ad infinitum such as:

Were both their ages measured in Earth years?

Are their ages given in binary?

And in the sentence "he found that if he added up all the birthdays he has had, including his age now, the result was one year more than his grandfather's age" do the words "he" and "his" refer to Joe or the Grandfather? It could have been Joe's grandfather comparing Joe's grandfather's age with Joe's great great grandfather.

Some tangents lead to insight. Some tangents show that some people have too much spare time.

It is clear from the question that we are talking of ages of Joe and his grandfather, there is no question of great grand father.

Dear klsraina,

If you read my statement carefully I wrote 'great great grandfather', not 'great grandfather' as you stated.

If you meant to say that it is clear that there is no great great grand father involved then you are wrong. There is abundant ambiguity because the words 'he' and 'his' could equally apply to Joe or to his grandfather. I've copied the question word for word below and edited out each occurance of 'he' and 'his':

Joe was thinking about his grandfather's age; Joe found that if Joe's Grandfather added up all the birthdays Joe's grandfather has had, including Joe's grandfather's age now, the result was one year more than Joe's grandfather's grandfather's age.

This make perfect grammatical sense, except for the clumsy repitition of the word grandfather rather than great great grandfather.

Had you said 'common sense would tell us that there is no great great grandfather involved' then I would have thanked you for agreeing with me.

You haven't disputed that the ages were both measured in earth years or that they were expressed in binary. Do you think those are worth debating? (See above reference to TMST)

Bravo Davo!

Some people just overthink every question. That is not thinking "outside the box". That is distorting and twisting and stretching and tearing and incinerating and....(well, you get the idea!)...the box.

Some people just have a need to show off how smart they think they are.

Oh, come ON, Tom!

Joe is 10 and grandpa is 19? Even if your logic of adding the number of birthdays (10 in your solution) and then adding Joes age (again 10) was correct (which it is NOT), how could a 19 year-old possibly be the grandfather of a 10 year old? And don't say he married the boy's grandmother, that only makes him a step-grandpa. It would be difficult (virtually impossible unless hormones kicked in reeeaaally early!) for a 19 year-old to have a 10 year-old son, let alone having a 10 year-old grandson!

The answer is : Boy is 10 and his grand father is 64.

Well done! That is the first correct answer I have seen. Apparently you are the only one to read the question carefully.

92,

Please say you are kidding and that was a tongue-in-cheek reply!

First of all, #90 was NOT the first one to give this WRONG answer. It was also given in #25 and #32 above.

Secondly, please read the problem carefully including the inferences. It does not say that Joe counts the number of his birthdays, then adds his current age. It says he adds up his birthdays (the ordinal numbers, 1st, 2nd, 3rd, etc. are implied here), and which includes his current age, the age of his last birthday, not adds his age.

By the way, see the answer given yesterday, March 6. 11 and 65 is the CORRECT answer!

Ok here goes nothing-

According to the text of the question...if joe adds up all the "birthdays" he has had (note-Birthday-anniversary of one's birth, and not to be confused with Birth Day - Day of one's birth), not the AGE at said birthdays - i.e. when he is 12 years old, he has had 12 birthdays - and includes his age now, he will end up with 2 X present age. If you subtract 1 from that number to get grandpa's age, and total those digits to arrive back at Joe's age, the only combination is that joe is 10, and grandpa is 19. Rather than sink to a geological or demographical slur, Joe and his grandfather are obviously clone and clone donor (once removed).

Ok here goes nothing- Boy, you got that right!

According to the text of the question...if joe adds up all the "birthdays" he has had (note-Birthday-anniversary of one's birth, and not to be confused with Birth Day - Day of one's birth), I am with you so far....

not the AGE at said birthdays - i.e. when he is 12 years old, he has had 12 birthdays - Ok, this is your mistake #1, the ordinal numbers, 1st, 2nd, 3rd, etc. are implied her. Most everyone else got it. So you add 1+2+3+4...

and includes his age now, he will end up with 2 X present age. Mistake #2: As the problem stated, includes his age now, not adds it in again!

If you subtract 1 from that number to get grandpa's age, and total those digits to arrive back at Joe's age, the only combination is that joe is 10, and grandpa is 19. Not just improbable, but impossible conclusion.

Rather than sink to a geological or demographical slur, I assume you are referring to my earlier post regarding inner cities and Arkansas, so as long as you are picking nits with the language of the challenge question, where is the geological slur? I assume you meant geographical slur, ....or was that demological?

Joe and his grandfather are obviously clone and clone donor (once removed). Boy, I hope this is meant tongue-in-cheek, since we are not talking Science Fiction here. Science Fact is that there has been no successful human cloning to date. Oh, wait, of course! Joe and his grandfather are SHEEP!