BMAD
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On 12/12/2018 at 4:52 PM, CaptainEd said:
I question whether the information given about a figure is sufficient to uniquely determine the number of included squares:
If I understand right, these two figures have the same N and P, but don’t have the same number of squares.o o o o
o o o o o
o o o
Nails = 12, perimeter = 7+sqrt2 + sqrt5
Squares =5
o o o o
o o o o o
o o o
nails=12, perimeter = 7 + sqrt2 + sqrt5
Squares = 6So then if we know N and P we should be able to bound squares x by a two values. Are those values always consecutive?
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On 12/11/2018 at 9:00 PM, CaptainEd said:
While this doesn’t answer the question with randomly placed 1s, I think it does give an upper bound to the number of columns that can be removed.
What is the smallest number of remaining columns that would allow (a) the 30 rows to be distinct, (b) each row to have 5 1-bits, and (c) all remaining columns to be distinct?
7 columns: 7!/2!5! = 21, too few distinct rows
8 columns: 8!/3!5! = 56, that’s enough.The first 12 columns in each of the 30 rows should be one of these patterns:
11000 00000 00
01100 00000 00
00110 00000 00
00011 00000 00
00001 10000 0000000 11000 00
00000 01100 00
00000 00110 00
00000 00011 00
00000 00001 1000000 00000 11
10000 00000 01The last 8 columns should be chosen from the C(8,5) possible patterns.
The resulting matrix satisfies the requirements because:
(a) the last eight columns differentiate the 30 rows
(b) the first twelve columns are distinct by design
(c) the last eight columns are distinct because of the distinct patterns chosen from the C(8,5)
(d) all row sums are 7 before column removal, and 5 afterwardsI agree
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Find a function where the arc lenth and area between any two randomly defined points is the same. There are two.
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Say we have the function:
y=x^x^x^x^x.....
Find an x value for which the derivative of this function converges.
If you are really clever you'll find the interval that converges.
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What is (-1/2)! x (-1/2)!
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Write the complex form (a + bi) for:
Sqrt ( i )
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Find the Limit as n goes to infinity for:
(1^n + 2^n + 3^n + 4^n....+ n^n)
---------------------over---------------------
(n^1+ n^2 + n^3 + n^4 ... + n^n)
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On 11/12/2018 at 9:12 AM, Donald Cartmill said:
The area would be that of the original triangle,however that area would vary as determined by the length of the 3rd side
I agree that it will vary but we can bound the area.
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Suppose you have a triangle that has 2-1 inch lengths. Divide this triangle into half by drawing a line from vertex between the two identical sides, choose one of the sides randomly and shade it. The non-shaded side is cut in half again. Choose one of these sides randomly and cut it in half again shading one random piece. If this pattern of cut, shade, cut, cut, shade, cut, cut, shade cut, cut,.... was to be continued forever, what would be the area of the shaded region?
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Maybe one of you answered this question: :)
I should have included this pic though...
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Suppose i have a circle. I cut off its arcs such that it became the biggest possible square i could make from that circle. What's the ratio of the edge of the circle to the middle of the edge of the square (assume minimum length) to the radius of the circle.
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On 6/4/2018 at 6:00 AM, bonanova said:
This sounds a lot, but not exactly, like eliminating variables from sets of equations. Is that the idea?
yes.
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No they just need to be distinguishable.
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Imagine you have several distinguishable rows composed of several distinguishable columns
The intersection of the rows and columns either have a 1 or a 0.
Each row sums to the same value and the question is how many of the columns can you eliminate assuming the the 1's in each row are randomly distributed across the columns
Example, there are 30 rows and 20 columns with each row containing 7 randomly dispersed 1's. How many columns can be eliminated reducing the total in each row by no more than 2.
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On 5/25/2018 at 6:25 AM, bonanova said:
The OP does not say how many apples there are. It says the proportion that are poisonous.
Question: was that the intent?
I meant them to be fractions so five in each
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Mining gold in a particular region is hard work. The metal only appears in 1% of rocks in the mine. But your friend Old Joe created a detector he’s been perfecting for months and it is finally ready. To your astonishment it always detect gold if gold is present. Otherwise it will have a 90% accuracy rate in detecting that a particular rock does not have gold. Working with Old Joe, You guys scan a large rock and determine that it gives a positive result. In loading it up, Old Joe realizes that both of you can't fit into the vehicle. He offers to sell his share to you for $200. You know that a rock of gold that size is worth easily $1000. Is that a fair price? Assume the vehicle remains with the proper owner.
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There are two bowls that you and a challenger must eat from. After flipping a coin you were selected to pick the bowl that each would eat from. In the first bowl there are three out of five poisonous apples. In the second bowl, there are two out of five poisonous apples. Whoever eats from the first bowl must eat two apples at random from the bowl. Whoever eats from the second bowl must eat three random apples from the second bowl. Which bowl should you pick to eat?
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On 5/11/2018 at 2:03 PM, CaptainEd said:
Sorry, proof was right, but method for calculation was confusing
If I am not mistaken, you found a way to calculate the area every time; which is wonderful, However, I wanted to know if it could be extended to know the amount of nxn squares that were defined within the shape not the precise area.
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1 hour ago, bonanova said:
Doctoring the figure a bit (while I think about solving it.)
o - o - o - o - o
| /
o o o o
| /
o - o - oQuestion: You mention the number of nails (12) and also the number of nails in the perimeter (here that's 10.) So for this example, would we be given N=12 or N=10? I'm thinking the latter, but would like to confirm.
nails are 12
perimeter are the side lengths, 8+sqrt(2)
Forgive my english, i think i see the confusion. When I say the number of nails in the perimeter what I am really trying to say is the number of nails throughout the shape.
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8 hours ago, CaptainEd said:
We are not limited to convex figures, right?
Hmmm, I only considered convex figures when making this problem. Let us first solve the simple case (only convex) then we could consider the more complex case with the relaxed condition.
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Imagine a piece of plywood with an array of evenly spaced nails forming small squares and consider that each square has side lengths of 1 unit. A simple closed shape is formed with a rubber band. If you knew the number of nails used in the perimeter and the perimeter itself, how could you predict how many squares can be counted inside this rubber band shape?
for example say the rubber band shape is outlining these nails:
* - * - * - * - *
| /
* * * *
| /
* - * - *
Perimeter = 8 + 2*sqrt(2)
Nails = 12
Squares: 6 squares (5 - 1x1 and 1 - 2x2) --- the result of 6, at a minimum, is what we are trying to predict.
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On 4/14/2018 at 3:11 AM, bonanova said:
Circle: Area = pi r2 Circumference = 2 pi r
Rectangle: Area = ab = pi r2 Perimeter = 2 (a + b) = 2 pi r Diagonal = d = sqrt (a2 + b2)1/2 Width = b.2 (a + b) = 2 pi r => (a + b) = pi r => (a + b)2 = pi2 r2.
[1] (a2 + 2ab + b2) = pi2 r2.
ab = pi r2 =>
[2] 2ab = 2 pi r2
[2] - [1] (a2 + b2) = pi2 r2 - 2 pi r2 = pi r2 (pi - 2) = ab (pi - 2) =>
[3] a2 - (pi - 2) b a + b2 = 0.
a = { (pi - 2) b +/- [ (pi - 2)2 b2 - 4b4 ]1/2 } / 2 = f (b)
Ratio of circumference to diagonal = g.
Write g in terms of rectangle width b.
g = 2 pi r / d = 2 (a + b) / ( a2 + b2 )1/2
g(b) = 2 ( f (b) + b ) / ( f2 (b) + b2 )1/2
excellent work as always bonanova but should
Spoiler{ (pi - 2) b +/- [ (pi - 2)2 b2 - 4b4 ]1/2 } / 2 actually be { (pi - 2) b +/- [ (pi - 2)2 b2 - 4b2 ]1/2 } / 2
Squares on a plywood
in New Logic/Math Puzzles
Posted
All you showed is the bound is wider than one thought.