09 May 22
A fort is a convex polygon 700m in perimeter.Patrols walk around the fort staying exactly 10m from it's walls.How far does the patrol walk in one circuit ?
Good luck.I don't even know what a convex polygon is!
My first question would be are they walking inside or outside,but as the question doesn't say,I suspect it doesn't matter!
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09 May 22
@venda saidIf it were inside, the question wouldn't be answerable. (Consider a fort 10m wide and 70m long...)
A fort is a convex polygon 700m in perimeter.Patrols walk around the fort staying exactly 10m from it's walls.How far does the patrol walk in one circuit ?
Good luck.I don't even know what a convex polygon is!
My first question would be are they walking inside or outside,but as the question doesn't say,I suspect it doesn't matter!
(A convex polygon, by the way, is a rectangle, or a triangle, or any multi-lined shape with only external angles. A star, for instance, is not one - it has some reflex angles and is therefore concave.)
Anyway, the answer is
762.83 and a bit metres
.
That's because, if you make a picture, you wil see that
they always walk along the sides for exactly the distance of that side. Those parts, therefore, add up to 700 as well. And when they get to a corner, they have to turn through a circle segment until they can start walking along the next side. But, because the fort is convex, and they end up back at the starting point for the next circuit, all those segments fit together to make exactly one circle of radius 10 - a length of 2 × pi × 10 = 62.83 plus change.
09 May 22
The paper doesn't give a numerical answer but there is an explanation:-
(700 + 20pi)m - My computer doesn't have the pi symbol
The path proceeds in straight lines that are parallel and equal in length to the walls interleaved with arcs of a circle 10m in radius.The arcs combine to make a full circle
The path proceeds in straight lines that are parallel and equal in length to the walls interleaved with arcs of a circle 10m in radius.The arcs combine to make a full circle
Way beyond me I'm afraid.
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09 May 22
@venda saidDraw it. Start by looking up what a convex polygon is, draw some (start with a rectangle, it's easiest to think about), then draw how you would walk a fixed distance around it. It's easier to see than to talk about.
The paper doesn't give a numerical answer but there is an explanation:-
Hidden content removed
Way beyond me I'm afraid.
25 May 22
@venda saidI can do it in...
A chess variant is played on a large board but the knights L shaped move is 7 squares in one direction and 10 in another.If I want to move a knight one square horizontally,what are the fewest moves that I must make ?
17 moves.
Reasoning:
No matter how we slice it, we are always setting some multiple of 7 in opposition to some multiple of 10. One way to get a net change of one square is 3*7 - 2*10. Another is 5*10 - 7*7. The number of sevens always seems to be odd.
Thus, to make the other net change of 0 squares, the number of tens must always be odd. The lowest multiple of 7 that cancels out an odd multiple of 10 is 70. Seven moves of 10 each, and ten moves of 7 each is 17 total moves.
Going back to the delta of 1 square, this means ten moves of 10 each, and seven of 7 each. I can cancel out as many pairs of 10 and 7 as needed, by moving the knight in opposite directions. This means I can make two net moves of 10 each, and three net moves of 7 each. Problem solved.
13 Jun 22
A railway system has 700 stations connected in a circle so that sometimes travellers have to call at 350 stations.New high speed services are proposed by the government that directly links stations 25 places apart,reducing the number of stations that have to be visited.The opposition argues that stations 28 apart should be linked instead.
Which proposal leads to the fewest visits in the worst case
13 Jun 22
2 edits
@venda saidThe worst point for any system of direct links is the one that is just shy of 350, halfway between the direct links.
A railway system has 700 stations connected in a circle so that sometimes travellers have to call at 350 stations.New high speed services are proposed by the government that directly links stations 25 places apart,reducing the number of stations that have to be visited.The opposition argues that stations 28 apart should be linked instead.
Which proposal leads to the fewest visits in the worst case
With direct links of 25, the worst case is
point #338, which requires 13 direct links plus 13 individual stops, for a total of 26 stations visited.
With direct links of 28, the worst case is
point #350, which requires 12 direct links plus 14 individual stops, for a total of 26 stations visited.
Given that
the worst-case number of stops is the same
, the government should save taxpayers some money and do the links 28 stations apart.
Edit: the solution may well be wrong, because I forgot to consider
starting out midway between the direct links
.
Edit: I think the given solution is OK. All that changes is that we do the small steps at the beginning of the trip, rather than the end.
23 Jun 22
2 edits
@venda said
I asked lots of people what their favourite season is:
16 didn't choose summer,14 didn't choose Autumn,15 didn't choose winter and 12 didn't choose spring.
How many people did I ask ?
We have six equations with 5 unknowns:
x=a+b+c+d
x=16+a
x=14+b
x=15+c
x=12+d
b+c+d=16
We can transform the last three equations thus:
b=a+2
c=a+1
d= a+4
After substituting b,c and d in equation 1 we get x=a+a+2+a+1+a+4 or x=4a+7
and substituting in equation 6 we get 16= a+2+a+1+a+4 or 16=3a+7
leading to a=3 and x=19.
So you asked 19 people, 3 choose summer, 5 choose Autumn, 4 choose winter and 7 choose spring.
x=a+b+c+d
x=16+a
x=14+b
x=15+c
x=12+d
b+c+d=16
We can transform the last three equations thus:
b=a+2
c=a+1
d= a+4
After substituting b,c and d in equation 1 we get x=a+a+2+a+1+a+4 or x=4a+7
and substituting in equation 6 we get 16= a+2+a+1+a+4 or 16=3a+7
leading to a=3 and x=19.
So you asked 19 people, 3 choose summer, 5 choose Autumn, 4 choose winter and 7 choose spring.