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Occasional puzzles for the bored


Nippy
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This is from Imperial College jizz mag:

A Swiss bank account number is between 30 and 80 digits and doesn't start with a zero, but the owner can't remember it. He can recall that multiplying it by 9 has the same effect as taking the last place digit and moving it to the front (eg so 134689 becomes 913468).

How long is the bank account number, and what is the number?

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32 minutes ago, Nippy said:

number is between 30 and 80 digits and doesn't start with a zero, but the owner can't remember it. He can recall that multiplying it by 9 has the same effect as taking the last place digit and moving it to the front

Interestingly that's also Ferguson's COVID forecast formula

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A mate wrote me from jail asking for some good puzzles. The one I sent him got a lot of interest. Many of the inmates and a few of the guards got involved. They solved it with 3 months of collective effort. I was later told that the correct formula for solving it is part of a 2nd years maths degree.

To calculate the electrical resistance of 2 resistors in series, it's R1 + R2 = R3. 

If the resistors are in parallel, it's 1/R1 + 1/R2 = 1/R3.

The puzzle:

Twelve 1k ohm resistors are soldered together to form a cube. What is the resistance between two diametrically opposing corners?

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Happy Renting
3 hours ago, The XYY Man said:

0.8333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333 Ohms

Or 5/6ths of an Ohm in old money....

 

XYY

 

Bollocks. It's 833.33333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333 Ohms

or 5/6th of a kiloohm.

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Happy Renting
4 hours ago, Nippy said:

This is from Imperial College jizz mag:

A Swiss bank account number is between 30 and 80 digits and doesn't start with a zero, but the owner can't remember it. He can recall that multiplying it by 9 has the same effect as taking the last place digit and moving it to the front (eg so 134689 becomes 913468).

How long is the bank account number, and what is the number?

Got it. Now what is his PIN?

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1 minute ago, The XYY Man said:

Very similar to the puzzle of tackling a woman, only much more predictable...

 

 XYY

I may have forgotten that too.

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One percent
1 hour ago, MrPin said:

I looked it up. There are several ways to tackle the resistor cube puzzle.

Is that similar to rubik’s cube?  

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9 hours ago, jm51 said:

The puzzle:
Twelve 1k ohm resistors are soldered together to form a cube. What is the resistance between two diametrically opposing corners?

What's the resistance across opposing corners of the cubic unit cell when the unit cell is in an infinite 3D lattice of 1k resistors?

Edited by Nippy
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5 hours ago, Nippy said:

What's the resistance across opposing corners of the cubic unit cell when the unit cell is in an infinite 3D lattice of 1k resistors?

Out of curiosity, I looked it up, and didn't understand it. It's ages since I dealt with that stuff.

Why are dusters yellow?

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17 hours ago, Nippy said:

This is from Imperial College jizz mag:

A Swiss bank account number is between 30 and 80 digits and doesn't start with a zero, but the owner can't remember it. He can recall that multiplying it by 9 has the same effect as taking the last place digit and moving it to the front (eg so 134689 becomes 913468).

How long is the bank account number, and what is the number?

Well I've taken it to:

First digit must be 1 (as the number gets no longer when multiplying it by 9), therefore last digit must be 9, x is the number, b is the length of the number:

9x = x/10 - 0.9 + 9 * 10^(b-1)

Resolving to, for x:

x = ((9 * 10^b) - 9)/89   where x is a whole number and b the number of digits but my spreadsheet is insufficiently accurate to identify where x is a whole number for 30 < b < 80.

I assume that there is a neater way.

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52 minutes ago, Frank Hovis said:

9x = x/10 - 0.9 + 9 * 10^(b-1)

The answer does appear to fulfill your equality expression above. There are arbitrary precision calculators around if you want to pursue this route?

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1 hour ago, Frank Hovis said:

x = ((9 * 10^b) - 9)/89   where x is a whole number and b the number of digits but my spreadsheet is

You know b is not a big number from the question, so you can find that out by doing some approximations. You know log10(x) and b are quite close in size by inspecting the formula you have for x.

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Happy Renting
On 10/08/2020 at 00:38, Nippy said:

What's the resistance across opposing corners of the cubic unit cell when the unit cell is in an infinite 3D lattice of 1k resistors?

I guess it tends to zero ohms.

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Happy Renting
1 hour ago, Nippy said:

Why?

Because there will be an infinite number of resistors in parallel in the lattice, and an infinite number of nodes and resistances the current can pass through..

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24 minutes ago, Happy Renting said:

Because there will be an infinite number of resistors in parallel in the lattice, and an infinite number of nodes and resistances the current can pass through..

True, but it wont't go to zero resistance.

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Happy Renting
24 minutes ago, Nippy said:

 I'll award you an 'A' on the basis of your geography mock result from 1990, if you can show some working.

Each node is connected to the rest of the infinite network by 6x  1k resistors effectively in parallel = 166R. Two nodes = 333R. The rest of the infinite network connecting them has an impedance asymptotic to zero.

Edited by Happy Renting
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On 14/08/2020 at 18:36, Happy Renting said:

I guess it tends to zero ohms.

I’d guess the opposite actually that infinite resistors tend to infinity. Like water pipes with voltage as a pump the water wouldn’t, in reality, get pumped miles away even if it’s theoretically the path of least resistance it will still mostly go the quickest/easiest path. I’d guess some small effect from the infinite lattice.

The problem with these sorts of problems where infinity, is an unrealistic scenario, the theory might tend to unrealistic solutions.

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On 09/08/2020 at 14:35, Nippy said:

This is from Imperial College jizz mag:

A Swiss bank account number is between 30 and 80 digits and doesn't start with a zero, but the owner can't remember it. He can recall that multiplying it by 9 has the same effect as taking the last place digit and moving it to the front (eg so 134689 becomes 913468).

How long is the bank account number, and what is the number?

I looked up the solution in frustration and it’s a shame it doesn’t state in the problem it can be solved with just basic mental arithmetic. Not saying I would have managed it but I reached for algebra just on the assumption it was the best route and can see I’d never have got there as it needs a working knowledge of Fermat’s little theorem which I’m not sure anyone not immersed in university maths would have.

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