He tells you the story of how he met a young Math student named Jane in Cambridge a few weeks ago. They spent a great evening together and then spoke a few times over the phone. Two days ago Jane invited him to join her and her family for the Thanksgiving dinner but she mused that he should figure the address from a puzzle. She told him that she lives on the even side of the long Beacon street. There are six houses on her side of the street in her block, and the sum of their numbers is 9870. She lives in the house with the lowest number among them.

She told him to figure out the number and come around 4pm. It is 6pm now and he was knocking on every first even-numbered doorbell searching for her house.

Do you think you can you help him?

(this puzzle is adapted from the

*The Mensa Genius Quiz-a-day*book by A.F.Salny)

Answers accepted all day long on Friday, on our Family Puzzle Marathon. They will be hidden until Saturday morning (EST) and everyone who provided a correct solution will get a puzzle point.

## 16 comments:

The house number is 1640. There are 6 houses on the even side of the block n+(n+2)+(n+4)+(n+6)+(n+8)+(n+10)totaling 9870. 6n+30= 9870

6n=9840,n=1640. The first house number is 1640. He might have had better luck if he knocked on the doors or rang the doorbells instead of knocking on the doorbell.

The answer is 1640

x + (x+2) + (x+4) + (x+6) + (x+8) + (x+10)

6x + 30 = 9870

x=1640

There are a few pieces of information that are useful for this puzzle:

1. She lives on the even side of the block (therefore the houses on her block are even)

2. There are six houses on her block.

(this tells us the house numbers must be 'close' together)

First, we figure out 9870 / 6 = 1645

This is the average of the houses on her block; however, since they're all even, they can't all be 1645.

One must be 1644 and 1646

another must be 1642 and 1648

The last one must be 1640 and 1650

Therefore, the row of houses are:

1640, 1642, 1644, 1646, 1648, 1650

And since Jane said she lives in the lowest number then she must live in 1640.

Great puzzle!

First I divided 9870 by 6 to find the arithmetic mean of the 6 house numbers. The mean is 1645, which would be right in the middle of all 6 numbers.....therefore, since the #'s are all even, they are 1640, 1642, 1644, 1646, 1648, 1650. Her's is the lowest of the 6, which means her house # is 1640.

1640 is her house number.

I divided 9870 by 6 to give me 1645. Then I put three houses on either side of that number, since they are even they are 1640, 1642, 1644, 1646, 1648, and 1650.

Let the Girl's house number = x

There are 6 houses on her side of the street, which is the even side, and her house is the lowest number.

Therefore, the houses on her block are X, X+2, X+4, X+6, X+8, and X+10.

The sum of the house numbers is 9870, so:

(X+X+2+X+4+X+6+X+8+X+10) = 9870

6X+30 = 9870

6X = 9840

x = 1640

Therefore, the girl's house number is 1640... although I can't help him explain why he's tow hours late!

The first 2 digits of all the houses in MOST blocks, including these, will likely be the same.

9870/6=1645, so this should be the 1600 block,

(I don't yet see how it could be the 1500, nor 1700's.)

1600X 6 = 9600

Seeking six two-digit even numbers totaling exactly 270...

270/6 = 45,,,might be possible to calculate that.

NO, wait, we don't need all the numbers! Jane should be in the house nearest the corner, done. Tom

The sum of four consecutive even numbers is 9870.

he needs the lowest of 6 house numbers.

x= the lowest house number

x+2=the next

x+4=the next

x+6=the next

x+8=the next

x+10=the next

x+(x+2)+(x+4)+(x+6)+(x+8)+(x+10)=9870

6x+30=9870

6x=9870-30

6x=9840

x=9870/6 9870/6=1640

x=1640

Jane's house number is 1640

If the sum is 9870, their average is 1645. So all the house would be on that block. The lowest even number on that block would be 1640.

The six house numbers on Jane's side of the block are

x, x+2, x+4, x+6, x+8, and x+10.

She said they sum to 9870, so

x + x+2 + x+4 + x+6 + x+8 + x+10 = 9870

6*x + 30 = 9870

6*x = 9840

x = 1640

He should go to 1640 Beacon Street.

Barry D. G. sent me an email at 8:53pm on Friday with his answer:

My guess is 1640

Barry - please provide an explanation next time and Anonymous above - feel free to post your name below to get a puzzle point in our list on the left.

The answer is 1640 and Pat, Wang, Donna, Katrina, Dennis, Lynnet, Kalonni, kj and Barry D.G. get a puzzle point each.

Tom - what's up with your line of thought? You almost caught this answer, but seem to get distracted by the smell of your morning coffee or a Black Friday Victoria Secret commercial.

Tom says: I probably should have done my thinking and writing in Word, edited it, and then sent it. But I'll stick with it, it's a good answer, for the suitor does not need the house number nor does the puzzle ask for the number; he only needs the right block, it was "given" that the lowest address would be it.

Jane is a rascal and we hold little hope for the relationship.

Tom

Tom offers more ... it is late at night but bear with me. I'm not convinced 1640 is the number, necessarily. In my neighborhood, there are big skips between house numbers, not consecutive even numbers. How about,for instance:

let x be the last 2 digits of the address, and it MAY end in zero

x+4

x+6

x+10

x+20

x+50

x+60

6x + 150 = 270

x can be 20

no?

Tom again, more awake, sorry, oops, but still,

1604, 1606, 1652, 1658, 1668, 1682 = 9870

You're not wrong about the coffee, Maria, but hey.

House number for Jane and her Thanksgiving COULD be 1604. Likely there are other qualifying combinations, plenty.

Answer 1: Yes, I think I can help him, I think we can help him.

Answer 2: Yes, I/we CAN help him.

Answer 3: The correct block is the 1600s, and he knows which house it must be, because Jane told him.

Answer 4: The house number was not asked for, and is not knowable except perhaps by going there, or with a Google map. The puzzle did not ask for the number, or the flavor of the dessert, etc. Given that Maria lives in Mass, we are likely discussing the Beacon street in Boston?

Answer 5: Many towns, but not all, are laid out in a grid, with cross streets at every 100. Most of my town is like that, however my own street runs about 600 numbers (4908-5498) with no cross street at all. I don't know Beacon Street.

Answer 6: It's tempting to assume consecutive even numbers like x+2, x+4, but no that's quite rare in my experience.

Answer 7: We can calculate that the lowest house number on the block is not greater than 1640, and not smaller than 1600. Beyond that, there seems to be a large number of solutions that will add up to 270 (or 9870).

Thanks for these, Maria!

Tom - you are right. I did not explicitly state that the house numbers are consecutive, we all just assumed it. Here, in Boston area and specially around the Beacon street, real estate is so expensive that any gap in house numbers is used as an opportunity to squeeze in a backyard shed or a doghouse, plaster the number on it and sell it. So, any (1606, 1652) pair would be a major real estate opportunity already taken advantage of. But I think you deserve a point for standing up to yoir point.

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