From Newsgroup: rec.puzzles

A robot sits in the top-left corner square of a 4 x 4 grid of squares. It
can only move to the right or down, and it must reach the bottom-right
corner. How many different routes are possible?

https://ibb.co/N6pRVYgT

Note: I found the answer, but don't immediately see how that relates to a general solution involving factorials. A good explanation of that
relationship would be welcome.

Thanks,
--
David Entwistle
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From Newsgroup: rec.puzzles

David Entwistle:

A robot sits in the top-left corner square of a 4 x 4 grid of squares. It can only move to the right or down, and it must reach the bottom-right corner. How many different routes are possible?

Assuming that it is constrained to move 1 square at a time, then it must
take 3 steps down and 3 to the right, and these can be in any order.
The number of permutations of DDDRRR is 6!/(3!3!) = 20, giving that many possible routes.

If it is allowed to step 2 or 3 squares at a time, the routes are the
same (for example, DD replaces D and D), ao there are still 20 routes.

If it is allowed to travel in increments of any length, then the number
of possible routes is infinite.
--
Mark Brader, Toronto | Thus, "plain english" is the same as
msb@vex.net | "near-field spin". --Carl Ginnow

My text in this article is in the public domain.
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From Newsgroup: rec.puzzles


msb@vex.net (Mark Brader) posted:

David Entwistle:

A robot sits in the top-left corner square of a 4 x 4 grid of squares. It can only move to the right or down, and it must reach the bottom-right corner. How many different routes are possible?

Assuming that it is constrained to move 1 square at a time, then it must
take 3 steps down and 3 to the right, and these can be in any order.

I assume a "4 x 4 grid of SQUARES" has FOUR squares across and down;
i.e. is defined by a 5 x 5 grid of points. The answer is thus C(8,4) = 70,
the same as Mark derived but with different numbers.

The way David phrases his question, he may be looking for a way to short-circuit the counting function C(8,4) and go from the problem
statement directly to factorials. I don't know how.

Cheers,
James
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From Newsgroup: rec.puzzles

David Entwistle:

A robot sits in the top-left corner square of a 4 x 4 grid of squares. It >>> can only move to the right or down, and it must reach the bottom-right
corner. How many different routes are possible?

Mark Brader:

Assuming that it is constrained to move 1 square at a time, then it must
take 3 steps down and 3 to the right, and these can be in any order.

James Dow Allen:

I assume a "4 x 4 grid of SQUARES" has FOUR squares across and down;
i.e. is defined by a 5 x 5 grid of points.

Agreed, but if the robot sites *in* a certain square, I assume we are
to assume that it sits on the center of the square, and if it moves
in steps from one square to another, the centers are where it goes.

The way David phrases his question, he may be looking for a way to short-circuit the counting function C(8,4) and go from the problem
statement directly to factorials. I don't know how.

See my answer.
--
Mark Brader, Toronto | "You know you've made it when you have
msb@vex.net | a disease named after you." --Andrew Niccol

My text in this article is in the public domain.
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From Newsgroup: rec.puzzles

On Mon, 7 Sep 2026 04:02:19 -0000 (UTC), David Entwistle wrote:

Note: I found the answer, but don't immediately see how that relates to
a general solution involving factorials. A good explanation of that relationship would be welcome.

My thoughts ...
y thoughts ...
thoughts ...
thoughts ...
houghts ...
oughts ...
ughts ...
ghts ...
hts ...
ts ...
s ...
...
...
..
.

Thanks for the replies. My solution is sketched out here:

https://ibb.co/wrzSWXBw

The problem is easy to break down in to smaller parts. There is only one
way to reach any of the squares in the top row; and only one way to reach
any of the squares in the left column. There are two ways to reach cell
number 6, and three ways to reach cell number 7. It quickly becomes clear
that the number of possible routes, to any particular cell, is the sum of
the routes to the cells that can access that cell (that is the sum of the routes to the cell above and to the left). Working thought that, the
answer, how many routes between top-left and bottom-right is 20, or 70 if
you have the additional column and row.

The problem was posed by an AI LLM, which provided the solution provided
by Mark, based on factorials.

For the smaller grid, which requires six moves to reach the diagonally opposing corner, I can see intuitively that are 6! ways of ordering those moves, if each is considered unique. I haven't yet seen, intuitively, how having two distinct groups of similar moves, R and D, decreases the 6! =
720 to 20, but it clearly does.

The sequence of moves for different sized grids; 1, 2, 6, 20, 70,
252 ...is described in OEIS series A000984.

https://oeis.org/A000984

The above provides lots of interesting information about the Catalan
numbers and their relationship to Pascal's triangle. I'll read up on the binomial theorem, which I recall vaguely from school.

Best wishes,
--
David Entwistle
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From Newsgroup: rec.puzzles

David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> wrote or quoted:

A robot sits in the top-left corner square of a 4 x 4 grid of squares. It >can only move to the right or down, and it must reach the bottom-right >corner. How many different routes are possible?

SPOILER WARNING This is an attempt to write an answer.

The robot has to move three times right and three times down.
We can only vary the order of those six moves.

For six different object, there are 6! individual orders, but
now we identify three objects twice, so the number should be
divided twice by 3!. This would give 6!/3!/3! = 720/6/6 = 120/6
= 20.

Now, a small Python program to test this!

source code

import itertools
seen = set()
n = 0
for p in itertools.permutations( [ 1, 1, 1, 2, 2, 2 ]):
if not p in seen:
seen.add( p )
n += 1
print(n)

output

20

Disclaimer I made an error first writing "6!/(2*3!)" which
is 60, but when I then saw that my Python script printed "20",
I corrected this. But otherwise I wrote this post without
looking up anything, not even for writing the Python script.


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From Newsgroup: rec.puzzles

On Tue, 8 Sep 2026 08:46:53 -0000 (UTC), David Entwistle wrote:

I'll read up on the binomial theorem, which I recall vaguely from
school.

Chapter 3 'What Comes Next?' of 'The Book of Numbers' covers related
problems, covers quite some ground, but does provide the background.
--
David Entwistle
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