perplexus.info :: Numbers : West of Evan, East of Odin

West of Evan, East of Odin (Posted on 2022-06-29)

In " Last house on the street " as well as in Henry Ernest Dudeney's original problem, there was a street with consecutive house numbers all on the same side of the street.
Now we have a different street with houses numbered 1, 2, 3, ... on alternating sides of the street. As you travel down this street from West to East, #1 is the first house on your left, then #2 on your right and so on, consecutively with all the odd numbers on the left (North) side of the street; all the evens on the right (South).
Odin lives in one of the odd numbered houses, and his best pal, Evan in one of the even numbered ones. It is unknown whether the last house number is even or odd. It is a very, very long street (way more than 5 houses).

One morning in May, Odin and Evan were having coffee discussing their favorite topics of math puzzles and savant arithmetic abilities. Odin noted "the sum of odd house numbers west of me and also on my side of the street is the same as the sum of even house numbers to the east of you on your side of the street. West of Odin equals East of Evan."
Evan agreed this this was quite remarkable, but added "what is more unusual is that the same is true if you flip flop the words 'east' and 'west'. In other words: West of Evan equals East of Odin."

....... 1 3 5 7 9 11 .......
West ...................... East
........ 2 4 6 8 10 12 ........

By example, if their house numbers were #5 and #8, and the last house were #12, then 1+3 would have to equal 10+12; and 7+9+11 would have to equal 2+4+6 (so this is not a solution).

By September of the next year, after their street was lengthened (more than doubled) with the same house numbering scheme, Odin brought his new friend Odessa to meet Evan. Shaking hands, Odessa remarked: "Evan, now that the length of the street is much longer, the mathematical relationship of your house number with Odin's is no longer true. But the good news is, your house number now shares that same relationship with my house number. West of Odessa equals East of Evan; and West of Evan equals East of Odessa."

(Odessa moved into a new house on the odd/North side of the street well east of Odin's address.)

(1). What were the house numbers of Odin, Evan, as well as the last house number on that May morning?

(2). What were Odessa's address (and Evan's which remained unchanged) and the last house number when Odessa met Evan the next September?

(3). Is there a solution if the last house is #5 ?

Bonus info: Several years earlier, the last house number was less than half of Evan's current address. Odin lived where he lives now. His house number had the same mathematical relationship with a different even numbered house, where Evart lived.
(Bonus Question): Many years earlier, what were Evart's (and Odin's) house numbers and the last house number?

No Solution Yet Submitted by Larry

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computer solution

Comment 1 of 1

for highOdd=11:2:650

for odin=1:2:highOdd

for highEven=highOdd-1:2:highOdd+1

for evan=2:2:highEven

if sum(1:2:odin-2)==sum(evan+2:2:highEven)

if sum(2:2:evan-2)==sum(odin+2:2:highOdd)

disp([highOdd, highEven, odin, evan])

end

for evan=[114 560]

for highOdd=101:2:2999

for highEven=highOdd-1:2:highOdd+1

sume=sum(evan+2:2:highEven);

sumw=sum(2:2:evan-2);

for odessa=1:2:highOdd

if sum(1:2:odessa-1)==sume

if sum(odessa+2:2:highOdd)==sumw

disp([highOdd, highEven, odessa, evan])

end

for highOdd=5

for odin=1:2:highOdd

for highEven=highOdd-1

for evan=2:2:highEven

if sum(1:2:odin-2)==sum(evan+2:2:highEven)

if sum(2:2:evan-2)==sum(odin+2:2:highOdd)

disp([highOdd, highEven, odin, evan])

end

The second portion, with its restriction of Evan to either 114 or 560 was written after the results of the first portion were known.

The first group below represents the first situation, before the road extension:

Each is:

highOdd, highEven, odin, evan
   277      276     253   114
   613      614     253   560

After the extension, the following are the mathematical possibilities for Odessa

highOdd, highEven, odessa, evan
   
   278     277        253   114
   614     615        253   560
  1362    1361       1241   560

Now, assuming Odessa doesn't live with Odin, the last row is the one that applies. It also shows that before the extension, Evan lived at 560 rather than 114.

Before the road extension, the road ended with 613 and 614 at the end, and after the extension, at 1361 and 1362. Evan lives at 560, Odin at 253 and Odessa at 1241, probably about 5 or 6 miles away, assuming lots at 50 feet width, thus counting 25 feet per house number given that there are two numbers per 50 feet.

For part (3) we get

5 4 5 2

Meaning that Odin lived at the last house, #5, with two houses to his west (1 and 3, totalling 4) and zero houses to his east, while Evan would have to live at #2, with zero houses to his west and house #4 to his east.

Bonus: Evert lived at 114, which we had ruled out as Evan's house number, when Odin was already at 253, and the last house number was 277.

Posted by Charlie on 2022-06-29 12:20:36

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