Easy math puzzles

Quote: ThatDonGuy

Unique Solution

Label the rows A-E from top to bottom, and the columns 1-5 from left to right

Numbers in curly braces can be in any order

B1-D1 is {1 2 4}
C3-C5 is {3 4 5}
D4-D5 is {1 3}
E1-E3 is {2 4 5}

A1-A3 is {1 3 5} or {2 3 4}
A5-B5 is {1 4} or {2 3}
C2-D3 is {1 4 1} or {1 2 3}
D5-E5 is {1 5} or {2 4}

If A5-B5 is {4 1}, D5-E5 cannot be any valid pair
Therefore A5-B5 is {2 3} and D5-E5 is {1 5}, which means C5 = 4,
which mens C2-D3 cannot be {1 4 1}, so it must be {1 2 3}
C2 cannot be 3, so either D2 or D3 = 3, which means D4 cannot be 3,
which means D4 = 1 and D5 = 3
Since E1-E4 have 2, 3, 4, and 5 in them, E5 = 1, which means D5 = 5
Since D4 = 1, D2 and D3 cannot be 1, so C2 = 1
Since C2-C5 have 1, 3, 4, and 5 in them, C3 = 2
C4 cannot be 1, 2, 3, or 4, so it is 5, and C3 = 3
This means that A4-B4 are {2 4}, so B3 = 5
Also, since B2 = 4, B4 cannot be 4, so it is 2, and A4 = 4
B2 cannot be 1, 4, 5, or 2, so it is 3, and A5 = 2; also, B1 = 1, so D1 = 4
D3 cannot be 4, 1, 5, or 3, so it is 2, and D2 = 3
E3 cannot be 5, 3, 2, or 1, so it is 4, and A3 = 1
A2 cannot be 4, 1, 3, or 2, so it is 5, and A1 = 3, which means E1 = 5 and E2 = 2

Each row, from left to right:
3 5 1 4 2
1 4 5 2 3
2 1 3 5 4
4 3 2 1 5
5 2 4 3 1
The solution is unique

Joke

Quote: rsactuary

[\spoiler]

Quote: Ace2

There’s a new casino game that plays as follows.

It starts with a bin of 35 balls numbered 1-35. You pick one number and you win $100 if it’s drawn. Only one ball is ever drawn from the bin

Then there is a second drawing for which 2 x 32 balls are added to the bin. You pick a number 1-99 and win $100 if it’s drawn

Then there is a third drawing for which 3 x 32 balls are added to the bin. You pick a number 1-195 and win $100 if it’s drawn

And so on.

The drawings and ball additions continue for as long as you want to play. You get paid $100 for every win...no limit

If this game was priced fairly, how much would it cost to play it ?

Quote: unJon

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This looks like it’s the sum from N=1 to infinity of 100/(3+32N(N+1)/2). Wolfram alpha says that sums to 25/6(3pi -8). Or about $5.93654.

Quote: Ace2

That is the correct answer. However, “for full credit” I’d like to see an explanation why it’s the answer.

Quote: Ace2

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60^2 / 5 * 300 / 5,280 =~40.9 mph

Meme

Quote: ThatDonGuy

As soon as I figure out how the sum over all positive integers n of 1 / ((4n + 2)^2 - 1) = (1 - 1/3 + 1/5 - 1/7 + ...) / 2 - 1/3, I'll let you know.

Quote: Ace2

You’re on the right track.

π/4 = 1 - 1/3 + 1/5 - 1/7 ....

If you combine the terms in pairs, eliminating the negatives, you get 2/3 + 2/35 + 2/99 + 2/195...Then subtract the 2/3, then multiply by 50 to get:

100(1/35 + 1/99 + 1/195...) =
50(π/4 - 2/3) =~ $5.94

Quote: ThatDonGuy

Wizard, if I still have any beers on my tab, you can give one to Ace2 - I learned at the last Spring Fling I went to that I just can't taste beers very well (they all taste like alcohol).

Quote: Wizard

I'd be happy to enjoy a beer with either of you. Also, substitutions may be made for drinks of equal or lesser value ;-).

Quote: Ace2

All women within a mile of the bar will evacuate

Quote: Wizard

I am used to that. I like to say that I have a certain magnetism when it comes to the ladies. Unfortunately, it is the repelling kind.

Quote: Wizard

In a two-dimensional world, A 5' ladder rests against the wall in an alley as well as touches the corner of a 1'x1' box, in which both one corner of the box fits into the corner of the ground and wall in the alley.

Find x.

In all fairness, a decimal representation to six decimal points is all that I ask.

Quote: ThatDonGuy

Good, because I get a quartic equation solution

Label the vertices of the triangle A, B, C such that AB is the hypotenuse and AC is the vertical (i.e. AC has length X + 1).
Label the remaining vertices of the square D, E, F such that CDEF is the square (i.e. D is on AC, E is on AB, abd F is on BC).

ADE is similar to EFB, so BF/FE = ED / DA, which means BF = ED * FE / DA = 1 * 1 / X = 1 / X.
Pythagorean Theorem: (AB)^2 = (AC)^2 + (BC)^2
5^2 = (X + 1)^2 + ((X + 1) / X)^2
25 X^2 = X^2 (X + 1)^2 + (X + 1)^2 = (X^2 + 1) (X^2 + 2X + 1) = X^4 + 2 X^3 + 2 X^2 + 2 X + 1
X^4 + 2 X^3 - 23 X^2 + 2 X + 1 = 0

There is a "generic" solution for quartic equations, but I broke out LibreCalc and applied Newton-Raphson to get X = 0.227205.

I just realized...there are two possible answers; rotate the image 90% counterclockwise, then reflect it through the vertical axis.
The other answer = 1 / X = 3.838501

Quote: Gialmere

In this age of covid, a small group of worshippers don masks and form a prayer circle on the lawn beside their church.



If 8 people stand in a circle with six feet of required social distance between them (along a chord):

1) What is the circumference of the circle?

2) What is the diameter of the circle?

3) How far in a straight line (chord) is Person A from Person C?

4) The prayer circle is formed on a square lawn measuring 25x25 feet. What is the maximum number of people who could stand in the circle and still be 6 feet apart?

Quote: Gialmere

In this age of covid, a small group of worshippers don masks and form a prayer circle on the lawn beside their church.



If 8 people stand in a circle with six feet of required social distance between them (along a chord):

1) What is the circumference of the circle?

2) What is the diameter of the circle?

3) How far in a straight line (chord) is Person A from Person C?

4) The prayer circle is formed on a square lawn measuring 25x25 feet. What is the maximum number of people who could stand in the circle and still be 6 feet apart?

Quote: IAchance5

Been a while, so hopefully I'm thinking on the right lines.....

Show Spoiler

Since the chord between 2 people is 6, then the radius of circle will be 7.89ft (6/(2 sin(.39 radians))
1) C = 49.57ft
2) d = 15.78ft
3) AC cord = 11.15ft (2*7.89) sin(.785 radians)
4) 12 people at most.....if we went to 13 people then we would go slightly over the radius of 12.5.....12.62 would be the radius for 13 people

Quote: Wizard

Answers to 1-3

radius = 3/sin(22.5) = 7.839377789
1. 2*pi*radius = 6*pi/sin(22.5) = 49.2563
2. 2*radius = 6/sin(22.5) = 15.6788
3. sqrt(2)*radius = 3*sqrt(2)/sin(22.5) = 11.0866

If this are right, I'll think about #4.

Quote: ssho88

No 4

Show Spoiler

sinθ > 3/12.5
θ >13.8865
2θ > 27.773

Max people < 360/27.773
Max people < 12.962

Max people = 12


Again, I don't really understand what is "stand in the circle" .... I assumed stand along the circle circumference.

Incomplete Joke Meme

<Insert your own prayer circle punchline here>

Quote: Gialmere

4) The prayer circle is formed on a square lawn measuring 25x25 feet. What is the maximum number of people who could stand in the circle and still be 6 feet apart?

Quote: Wizard

You put $1 in a bank account at a fixed interest rate of 10%, compounded monthly. Over an infinite period of time, what ratio of the time will the balance start with a 1?

Quote: Wizard

You put $1 in a bank account at a fixed interest rate of 10%, compounded monthly. Over an infinite period of time, what ratio of the time will the balance start with a 1?

Quote: Wizard

You put $1 in a bank account at a fixed interest rate of 10%, compounded monthly. Over an infinite period of time, what ratio of the time will the balance start with a 1?

Quote: Ace2

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I guess log(2) =~ 30.1%

Quote: ThatDonGuy

Answer

Let I be the "interest multiplier" - the amount you multiply the principle by each compounding term.

If it takes A terms to get from 10^k to 2 x 10^k, then (10^k) x (I^A) = 2 x 10^k
I^A = 2
A log I =log 2
A = log 2 / log I

If it takes B terms to get from 2 x 10^k to 10^(k + 1), then (2 x 10^k) x (I^B) = 10 x 10^k
I^B = 5
B = log 5 / log I

The ratio is log 2 / (log 2 + log 5); using base 10 logarithms, log₁₀ 2 + log₁₀ 5 = 1, so the ratio is log₁₀ 2 = about 0.30103

Quote: charliepatrick

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Rather than look at states just look at all the ways to get there and how many of 7776 ways to throw five dice results in the required outcome.
For instance initially throwing 5 dice, you can get
6-only : 2101
6 and 5: 1320
6 5 and 4 : 1230
miss : 3125
Rinse and repeat for using four dice trying to get 5 4 and three dice to get 4.

Work out all ways to get passes and fails, check it adds up to 7776³.
Pass = 253 888 187 826
Fail = 216 296 796 750
Pr Pass = 53.9975%

Outcome	Roll 1	Roll 2	Roll 3	Perms 1	Perms 2	Perms 3	Perms
Pass	6 5 4	any	any	1 230	7 776	7 776	74 373 396 480
Pass	6 5	4	any	1 320	3 276	7 776	33 625 912 320
Pass	6 5	miss	4	1 320	4 500	3 276	19 459 440 000
Pass	6	5 4	any	2 101	1 812	7 776	29 603 325 312
Pass	6	miss	5 4	2 101	3 750	1 812	14 276 295 000
Pass	6	5	4	2 101	2 214	3 276	15 238 687 464
Pass	miss	6 5 4	any	3 125	1 230	7 776	29 889 000 000
Pass	miss	6 5	4	3 125	1 320	3 276	13 513 500 000
Pass	miss	6	5 4	3 125	2 101	1 812	11 896 912 500
Pass	miss	miss	6 5 4	3 125	3 125	1 230	12 011 718 750
Total Pass				0	0	0	253 888 187 826
Fail	6 5	miss	miss	1 320	4 500	4 500	26 730 000 000
Fail	6	5	miss	2 101	2 214	4 500	20 932 263 000
Fail	6	miss	5	2 101	3 750	2 214	17 443 552 500
Fail	6	miss	miss	2 101	3 750	3 750	29 545 312 500
Fail	miss	6 5	miss	3 125	1 320	4 500	18 562 500 000
Fail	miss	6	5	3 125	2 101	2 214	14 536 293 750
Fail	miss	6	miss	3 125	2 101	3 750	24 621 093 750
Fail	miss	miss	6 5	3 125	3 125	1 320	12 890 625 000
Fail	miss	miss	6	3 125	3 125	2 101	20 517 578 125
Fail	miss	miss	miss	3 125	3 125	3 125	30 517 578 125
Total FAIL				0	0	0	216 296 796 750

Quote: IAchance5

No Interest Theory formulas needed.....

Show Spoiler

1. calendar vendor $3.65 (days)
2. casino worker $7.11 (always open)
3. deer hunter $30.30 (shotgun)
4. electrician $1.20 (volts)
5. golf instructor $.18 (holes)
6. history teacher $14.92 (Columbus)
7. IRS agent. $10.40 (form)
8. jet pilot. $7.47 (plane)
9. mathematician $3.14 (Pi)
10. paramedic $9.11 (emergency)
11. piano tuner $.88 (keys)
12. soap maker. $99.44 (ivory pure)

Quote: Joeman

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1. calendar vendor --> e. $3.65
2. casino worker -----> f. $7.11
3. deer hunter ------> m. $30.30*
4. electrician -------> c. $1.20
5. golf instructor ---> a. $.18
6. history teacher --> j. $14.92
7. IRS agent. -------> i. $10.40
8. jet pilot. ---------> g. $7.47
9. mathematician --> d. $3.14
10. paramedic -----> h. $9.11
11. piano tuner ----> b. $.88
12. soap maker ----> n. $99.44

* You should have paid him $30.06 -- you'd have literally gotten more bang for your buck!

Quote: Gialmere

I asked my girlfriend what sort of books she's interested in.

She said check books.

Quote: gordonm888

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The digits must add up to 10. Thus:

0 1 2 3 4 5 6 7 8 9
6 2 1 0 0 0 1 0 0 0

Quote: charliepatrick

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There are a large number of zeroes, say N, so N is the first digit. There is (probably) only one of those (otherwise there would be more than 10 digits in the number), so there is a 1 under N. There is now at least one "1", so there is a number under 1, however it can't be 1 (as there would now be two of them), so it has to be 2 and a "1" under 2. From this there are six zeroes so the leading digit is 6.

6 2 1 0 0 0 1 0 0 0