From Newsgroup: rec.puzzles

A slight variation on the original puzzle by Dudeney.

Simon Short and Lucy Long struck up a friendship on-line after
discovering they had a common interest in puzzles and realizing that
they lived in the same town.

They had been corresponding by email, but decided it would be nice if
they could write to each other, for a more personal touch. As they
hadn't actually met, both wanted to be sure that the other was genuine
about their interest in puzzles. They agreed to set each other a puzzle
which would provide their address.

Simon emailed Lucy to explain that he lived on Short Street, which as
its name suggests, was short, having fewer than thirty houses. The
houses were numbered consecutively on his side of the street, one, two,
three and so on. Simon also explained that all the numbers on one side
of him added up exactly the same as all the numbers on the other side of
him.

Lucy replied to explain that that was a great coincidence. She lived on
Long Lane, which as its name suggests, was long, having more than thirty houses. The houses were numbered consecutively on her side of the
street, one, two, three and so on. Lucy also explained that all the
numbers on one side of her added up exactly the same as all the numbers
on the other side of her.

At what house numbers do Simon and Lucy live?

Any other variations on this theme would be welcome.
--
David Entwistle

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From Newsgroup: rec.puzzles

On 03/09/2025 13:10, David Entwistle wrote:

A slight variation on the original puzzle by Dudeney.

Simon Short and Lucy Long struck up a friendship on-line after
discovering they had a common interest in puzzles and realizing
that they lived in the same town.

They had been corresponding by email, but decided it would be
nice if they could write to each other, for a more personal
touch. As they hadn't actually met, both wanted to be sure that
the other was genuine about their interest in puzzles. They
agreed to set each other a puzzle which would provide their address.

Simon emailed Lucy to explain that he lived on Short Street,
which as its name suggests, was short, having fewer than thirty
houses. The houses were numbered consecutively on his side of the
street, one, two, three and so on. Simon also explained that all
the numbers on one side of him added up exactly the same as all
the numbers on the other side of him.

Lucy replied to explain that that was a great coincidence. She
lived on Long Lane, which as its name suggests, was long, having
more than thirty houses. The houses were numbered consecutively
on her side of the street, one, two, three and so on. Lucy also
explained that all the numbers on one side of her added up
exactly the same as all the numbers on the other side of her.

At what house numbers do Simon and Lucy live?

Any other variations on this theme would be welcome.

.....Spoiler: the product of the lowest primes greater than the
two lowest non-trivial answers is shown below.
.....Spoiler: the product of the lowest primes greater than the
two lowest non-trivial answers is shown below
.....Spoiler: the product of the lowest primes greater than the
two lowest non-trivial answers is shown belo
.....Spoiler: the product of the lowest primes greater than the
two lowest non-trivial answers is shown bel
.....Spoiler: the product of the lowest primes greater than the
two lowest non-trivial answers is shown be
.....Spoiler: the product of the lowest primes greater than the
two lowest non-trivial answers is shown b
.....Spoiler: the product of the lowest primes greater than the
two lowest non-trivial answers is shown
.....Spoiler: the product of the lowest primes greater than the
two lowest non-trivial answers is shown
.....Spoiler: the product of the lowest primes greater than the
two lowest non-trivial answers is show
.....Spoiler: the product of the lowest primes greater than the
two lowest non-trivial answers is sho
.....Spoiler: the product of the lowest primes greater than the
two lowest non-trivial answers is sh
.....Spoiler: the product of the lowest primes greater than the
two lowest non-trivial answers is s
.....Spoiler: the product of the lowest primes greater than the
two lowest non-trivial answers is
.....Spoiler: the product of the lowest primes greater than the
two lowest non-trivial answers is
.....Spoiler: the product of the lowest primes greater than the
two lowest non-trivial answers i
.....Spoiler: the product of the lowest primes greater than the
two lowest non-trivial answers
.....Spoiler: the product of the lowest primes greater than the
two lowest non-trivial answers
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two lowest non-trivial answer
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two lowest non-trivial answe
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two lowest non-trivial answ
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two lowest non-trivial ans
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two lowest non-trivial an
.....Spoiler: the product of the lowest primes greater than the
two lowest non-trivial a
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two lowest non-trivial
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two lowest non-trivial
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two lowest non-trivia
.....Spoiler: the product of the lowest primes greater than the
two lowest non-trivi
.....Spoiler: the product of the lowest primes greater than the
two lowest non-triv
.....Spoiler: the product of the lowest primes greater than the
two lowest non-tri
.....Spoiler: the product of the lowest primes greater than the
two lowest non-tr
.....Spoiler: the product of the lowest primes greater than the
two lowest non-t
.....Spoiler: the product of the lowest primes greater than the
two lowest non-
.....Spoiler: the product of the lowest primes greater than the
two lowest non
.....Spoiler: the product of the lowest primes greater than the
two lowest no
.....Spoiler: the product of the lowest primes greater than the
two lowest n
.....Spoiler: the product of the lowest primes greater than the
two lowest
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two lowest
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two lowes
.....Spoiler: the product of the lowest primes greater than the
two lowe
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two low
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two lo
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.....Spoiler: the product of the lowest primes greater tha
.....Spoiler: the product of the lowest primes greater th
.....Spoiler: the product of the lowest primes greater t
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.....Spoiler: the product of the lowest primes greater
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.....Spoiler: the product of the lowest primes great
.....Spoiler: the product of the lowest primes grea
.....Spoiler: the product of the lowest primes gre
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.....Spoiler: the product of the lowest primes g
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.....Spoiler: the product of the lowest pr
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.....Spoiler: the product of the lowest
.....Spoiler: the product of the lowes
.....Spoiler: the product of the lowe
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.....
....
...
..
.

I found several solutions, but I think the two you are after, if
you take the lowest primes that exceed them and multiply them
together, will give you 259.
--
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: rec.puzzles

On 03/09/2025 13:47, Richard Heathfield wrote:

.....Spoiler: the product of the lowest primes greater than the two
lowest non-trivial answers is shown below.

Well done - they are the two I was looking for. Was that solved
algebraically, or algorithmically?
--
David Entwistle
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: rec.puzzles

On 03/09/2025 15:39, David Entwistle wrote:

On 03/09/2025 13:47, Richard Heathfield wrote:

.....Spoiler: the product of the lowest primes greater than the
two lowest non-trivial answers is shown below.

Well done - they are the two I was looking for. Was that solved algebraically, or algorithmically?

.....not a long spoiler so beware - revealing solution method.
.....not a long spoiler so beware - revealing solution method
.....not a long spoiler so beware - revealing solution metho
.....not a long spoiler so beware - revealing solution meth
.....not a long spoiler so beware - revealing solution met
.....not a long spoiler so beware - revealing solution me
.....not a long spoiler so beware - revealing solution m
.....not a long spoiler so beware - revealing solution
.....not a long spoiler so beware - revealing solution
.....not a long spoiler so beware - revealing solutio
.....not a long spoiler so beware - revealing soluti
.....not a long spoiler so beware - revealing solut
.....not a long spoiler so beware - revealing solu
.....not a long spoiler so beware - revealing sol
.....not a long spoiler so beware - revealing so
.....not a long spoiler so beware - revealing s
.....not a long spoiler so beware - revealing
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.....not a long spoiler so beware - reveali
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.....not a long spoiler so beware - reve
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.....not a long spoiler so beware - re
.....not a long spoiler so beware - r
.....not a long spoiler so beware -
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.....not a long spoiler so be
.....not a long spoiler so b
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.....not a long spoiler s
.....not a long spoiler
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.....not a long spoile
.....not a long spoil
.....not a long spoi
.....not a long spo
.....not a long sp
.....not a long s
.....not a long
.....not a long
.....not a lon
.....not a lo
.....not a l
.....not a
.....not a
.....not
.....not
.....no
.....n
.....
....
...
..
.

Kinda both.

From any candidate address h you can calculate the street length
s like this:

r = h - 1

Total to right = r*(r+1)/2
Total to left = r*(r+1)/2

Total to left and right t = r*(r+1)=h(h-1)

Total including Simon's (or Lucy's) house number:-

h(h-1)+h = h^2

s = (sqrt(h^2*8+1)-1)/2

That's just n(n+1)/2 on its head.

The point being that h^2*8+1 has to be an integer.

Having got thus far, I figured I deserved a pat on the CPU, so I
reached for my compiler.
--
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: rec.puzzles

In article <1099b6p$14jd4$1@dont-email.me>,
David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> wrote:

A slight variation on the original puzzle by Dudeney.

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Consider house m in a street of n houses.

The sum of the smaller numbers is m(m-1)/2, the sum of the larger ones
is n(n+1)/2 - m(m+1)/2. These are equal if and only if

m^2 - m = n^2 + n - m^2 - m
<=> 2 m^2 = n^2 + n
<=> m^2 = n(n+1)/2

That is, if and only if the square of m is a triangular number.

https://oeis.org/A001109

-- Richard
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From Newsgroup: rec.puzzles

On Wed, 3 Sep 2025 17:00:33 -0000 (UTC), richard@cogsci.ed.ac.uk
(Richard Tobin) wrote:

In article <1099b6p$14jd4$1@dont-email.me>,
David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> wrote:

A slight variation on the original puzzle by Dudeney.

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Consider house m in a street of n houses.

The sum of the smaller numbers is m(m-1)/2, the sum of the larger ones
is n(n+1)/2 - m(m+1)/2. These are equal if and only if

m^2 - m = n^2 + n - m^2 - m
<=> 2 m^2 = n^2 + n
<=> m^2 = n(n+1)/2

That is, if and only if the square of m is a triangular number.

https://oeis.org/A001109

-- Richard

I vaguely remembeed reading that Euler had solved this
problem when I was on the track of the slightly different
kind of partition: for what values of N does there exist
an m such that

1 + 2 + 3 + ....... + n = (n+1) +( n+2) + ..... + N

But, I could find nothing in my notes about Euler's work.
But, googling, provided the link in no time.

https://en.wikipedia.org/wiki/Square_triangular_number

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From Newsgroup: rec.puzzles

On 03/09/2025 13:10, David Entwistle wrote:

At what house numbers do Simon and Lucy live?

Once you've arrived at your solution, then if you have 50 minutes to
spare, you may enjoy the Mathologer video describing Ramanujan's partial fraction solution to the original problem.

https://www.youtube.com/watch?v=V2BybLCmUzs
--
David Entwistle
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: rec.puzzles

In article <109bcj0$1k0du$1@dont-email.me>,
David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> wrote:

Once you've arrived at your solution, then if you have 50 minutes to
spare, you may enjoy the Mathologer video describing Ramanujan's partial >fraction solution to the original problem.

I don't have 50 minutes right now, but I think the "partial fractions"
here are what are usually called the "convergents" of a continued
fraction.

-- Richard
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: rec.puzzles

On 04/09/2025 10:56, Richard Tobin wrote:

I don't have 50 minutes right now, but I think the "partial fractions"
here are what are usually called the "convergents" of a continued
fraction.

Yes, having looked at the link below, that looks to be what I should
have said. Thanks for the correction.

https://en.wikipedia.org/wiki/Continued_fraction

NRICH provides some information if continued fractions are new to you.

https://nrich.maths.org/tags/continued-fractions
--
David Entwistle
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