A Fistful of Brain Teasers

YYURYYUB.

Can you complete the sentence?

TOBYJUG posted:

YYURYYUB.

Can you complete the sentence?

Wow, takes me back! ICUR ...

.......YY4me

Nice one Toby.

Ok, i’m sat in a restaurant in Golden waiting for the server to take our order. We already have a glass of water each but Mrs D doesn’t like ice and each glass had loads of ice.

So she has just carefully fished most of the ice out of her glass and carefully put it into mine. The water level is “just” level with the brim of the glass and the floating ice looks like icebergs above the brim.

I have decided to leave my glass of ice/water (the wine has arrived) to see what happens as the ice melts.....

........any predictions ?

Archimedes Principle applies and the level does not change

Funny, my partner brought up a large glass of water filled up to the brim with lots of ice up at bedtime the other month when it was super hot. She said I should drink it before the ice melted and it would overflow onto the sideboard.   Apart from laughing I couldn't think of an easy way to explain this not happening.

Water expands when it freezes and contracts when it melts, therefore I would expect the total volume of water in your glass to fall slightly as the ice melts. However, as the ice is less dense than the water, some portion of the cubes will be above the fill level at the beginning, so the glass would start off in an “overfilled” state.

I expect that these would cancel each other out, and that the top level of the water would remain unchanged. However, evaporation would be taking place during this time (albeit curtailed by the presence of the ice) so I’d expect the fill level to have fallen very slightly by the time all the ice has melted.

Don Atkinson posted:

Ok, i’m sat in a restaurant in Golden waiting for the server to take our order. We already have a glass of water each but Mrs D doesn’t like ice and each glass had loads of ice.

So she has just carefully fished most of the ice out of her glass and carefully put it into mine. The water level is “just” level with the brim of the glass and the floating ice looks like icebergs above the brim.

I have decided to leave my glass of ice/water (the wine has arrived) to see what happens as the ice melts.....

........any predictions ?

I read an article by an American professor once who used this example to explain why melting Ice bergs will not cause sea levels to rise. I have no idea if that is correct or not but in the case of iced water the level doesn't rise.

Bob the Builder posted:

I read an article by an American professor once who used this example to explain why melting Ice bergs will not cause sea levels to rise. 

Sounds like the kind of professor Trump would be keen to hire to head the US EPA.

Bob the Builder posted:

I read an article by an American professor once who used this example to explain why melting Ice bergs will not cause sea levels to rise. I have no idea if that is correct or not but in the case of iced water the level doesn't rise.

Melting icebergs won't cause a rise in sea levels as they already have a displacement equal to their own mass.

Melting of the Greenland and Southern Polar ice caps (which are supported by land below) will cause a rise in sea level.

Whilst Archimedes’ Principle does indeed apply, as noted above, the mass of water+ ice being the same as the mass of water once the ice has melted, the reality  is a bit more sublte and depends on other, unknown, factors:  

Water is at its densest at 4ºC. If the water is alll at that temperature, and the ice has come straight out of a freezer at, say, -18ºC, as the ice is warming up to zero before melting it is cooling the water further, so the water level will rise. If, however, the water was above 4º then it would contract and the level would fall until or unless the amountbof ice below zero cools the water down to below 4º.

But this is all ignoring the room temperature: if the room temperature is above that of the water, it will be heating the water while the ice is cooling it. The rate of heating from air through the glass might be expected to be lower than the rate of cooling by the ice, so the water would still cool down, with the aforementioned effects, however if the restaurant is in a tropical location and has no air conditioning that may not necessarily be the case, and it is possible that the water temperature could rise - and certainly will rise after the ice has finished melting, when once above 4ºC it will start expanding, and once above the original temperature the level will rise above the original level. (The last assuming that the water was originally above 4ºC otherwise once the water goes above whatever temperature has the same density as the water’s starting temperature .)

Now, this leads to a second question: in those water level rising scenarios, would the glass necessarily overflow if the water level starts level with the brim of the glass as Don described?

And you are all overlooking surface tension effects. Don't over-complicate.

Simple. As the ice melts, you drink your wine.

Yes, the temperature outside last night was 4 deg C. However, in the restaurant it was 21 deg C.

The water had been chilled to 4 deg C and the ice from the freezer was a cool -18 deg C

The relative mass of water was 4 parts and ice was 1 part.

Just to simplify things, we ate our meal quickly and the ice had all melted by the end of the meal so you can assume that all changes were adiabatic.............

..........we are now enjoying breakfast, the temperature outside is a cool -4 deg C and.................

Don Atkinson posted:

Ok, i’m sat in a restaurant....

........any predictions ?

Yes. I'd be happier with I'm sitting   ;-)

Now, the square root of 1764 is dead easy to find when you have your Excel spreadsheet to hand or that old pocket calculator, but..............

.......who can recall how to find these  square roots the good old fashioned way using pencil and paper ?

....and of course, describe the method !

Perhaps you could even explain why the method works ?

If 1764 is too easy, try 1784 instead, to three decimal places............

Don Atkinson posted:

Now, the square root of 1764 is dead easy to find when you have your Excel spreadsheet to hand or that old pocket calculator, but..............

.......who can recall how to find these  square roots the good old fashioned way using pencil and paper ?

....and of course, describe the method !

Perhaps you could even explain why the method works ?

If 1764 is too easy, try 1784 instead, to three decimal places............

For 1764, visual inspection  gets you there. 1600 is 40 squared. So we need to find 164 more. 164 is approx 4*40, so we’re looking at approx 40+4/2, try 42 and it turns out to be right. 

Using the Long division method

We know that the closest whole number square to 1784 is 42 i.e. 42^2 is 1764 so our root of 1784 will be 42.xyz.

Dividing 1784 by 42 leaves a remainder of 20 to which we add 2 zeros i.e. 2000. Doubling our quotient (42) produces 84 so we are now looking for a divisor of 84x which when multiplied by x gets us as close to but below 2000.

3 multiplied by 843 will be too large so we take 2 multiplied by 842 i.e. 1684. This leaves a remainder of 316 from 2000 and our root becomes 42.2yz. Repeating the process by adding a further 2 zeros to the remainder i.e. 31600 and doubling the root quotient i.e. 42.2 becomes 84.4 so we need to find 844y such that 844y multiplied by y gets us as close to but below 31600. 4 by 8,444 is too great so our root becomes 42.23. 3 times 8,443 is 25,329 leaving a remainder of 6,271 from 31,600.

Using one more iteration the final figure to 3 decimal places is 42.237 which produces a square of 1783.964.     

Hi SJ,      

Looks like you learned the same system as I did !

Here is my example of the "long division" method that I recall from school. BTW, the "pairs" should be created starting at the decimal point and working left for the integers. If there's an odd number of integers then the first digit of the solution can only be 1, 2, or 3.

For all of us "over-60s"...................

This, and the following sequence of posts might bring back "happy days"........(or not !)

4 is the largest integer root of 18

4 x 4 = 16

18 - 16 = 2

4 + 4 = 8

So far, so good ?

Bring down the next "pair" of numbers. in this case "00"

Easy !

This needs a bit of "thinking" or "trial & Error" !

We need a single digit to replace all three "*", such that 8* x * is the biggest number that is equal to, or less than 200.

81 x 1 = 81  (ok but probably not the biggest number)

82 x 2 = 164 (ok, and might be the biggest number)

83 x 3 = 249 (too big, so let's go back to 82 x 2 = 164)