Brain Teasers ? or 50 Years On........... ?

Excel has quite few quirks...............

When I was at school we used to be asked for answers (say) to 2dp or 3 sfg. In order to achieve these levels of accuracy, it was necessary to work to 3dp or 4 sfg.

So with Excel I need an answer to 1dp. I type into cell B3 17.45 and into cell B4 18.36. In cell B5 I put the usual "=sum(B3:B4)" and hey presto 35.81.

But I only want the answer to 1dp so I reset the format to  display 1dp and hey presto............I see 17.5 + 18.4 = 35.8.

Something isn't right !!

Now, HH who is an accountant, would have a fit if I printed this out and told him that his wages for Thursday and Friday were £35.8 rather than.............

I should have added, that to be consistent with school days, I should have entered the input into cells formatted for 2dp and only re-formatted the answer cell to 1dp.

But I am continually surprised at how many people produce documents such as my initial post on this subject

Hi Don,

Until your follow up post, I was trying to work out what you thought actually should happen!

It's a good thing that HH is an accountant not a banker, otherwise he'd point out that for financial calculations EU rules require them to be performed so as to ensure a precision of 4 significant digits after the decimal point (normally 6 dp integers are used).

The other thing people forget about excel is that it uses IEEE 64bit float format so it's calculations aren't exact enough for large financials anyway.

I might point out that it's should be its, but I won't. Oops. 

Hungryhalibut posted:

I might point out that it's should be its, but I won't. Oops. 

Dyslexia Rules KO!

What about the dyslexic diabolist who sold his soul to Santa?

A sewer worker is inside a large diameter, circular sewer, ie much larger than the worker. The sewer surface is smooth and frictionless

The worker wants to inspect the ceiling and has a ladder whose length equals the diameter of the sewer.

He mounts the ladder and continues to climb until he reaches the other end of the ladder.

What happens ?

I assume you mean tubular, rather than circular.

The sewage worker 'has' the ladder, therefore it isn't already erected in the sewer.

As the Ladder is the same length as the diameter of the sewer but has finite width:
1 he can't erect it completely vertically in the sewer.
2 he can't align it onto a diameter of the pipe, as he can't turn it (it's diagonal is greater than it's height).

The nearest he can achieve is...
He lies on his back under the ladder and starts at one end of the ladder holding it above himself.  By moving the ladder along above himself (rung by rung) and increasing the angle he may be able to get the ladder to the ceiling (this depends on relative moment, due to weights and lengths).  Even if he does, the ladder won't stay put as it's the length of the diameter but has width, so can't be erected vertically and due to the frictionless surface, can't wedge the ladder.

Therefore he can't climb the ladder.

Further to the lorry fuel consumption problem I worked it in Mathematica for 'fun' and for interest show some of the working:

lorrydata references the data in table form.

Fit[lorrydata, {1, x, x^2}, x]     - the Fit function does the least squares stuff and returns the expression:

20.483 - 2.67236 x + 0.114133 x^2

 This expression plotted over the data points to see the good fit.

Then declare a function of the above expression:

f[x_] := 20.48 - (2.67 x) + (0.11 (x^2))

and solve for the load

f[7.6]

answer:  Fuel 6.5416   (km/litre) 

Beats the hell out of excel.

Huge posted:

I assume you mean tubular, rather than circular.

 Ah, I meant "uniform circular cross-section" ergo "tubular"

The sewage worker 'has' the ladder, therefore it isn't already erected in the sewer.

The ladder is already erected in the sewer. Assume it was lowered down an access shaft, placed more or less upright, then slid into the "tubular" part of the sewer.

As the Ladder is the same length as the diameter of the sewer but has finite width:
1 he can't erect it completely vertically in the sewer.
2 he can't align it onto a diameter of the pipe, as he can't turn it (it's diagonal is greater than it's height).

The nearest he can achieve is...
He lies on his back under the ladder and starts at one end of the ladder holding it above himself.  By moving the ladder along above himself (rung by rung) and increasing the angle he may be able to get the ladder to the ceiling (this depends on relative moment, due to weights and lengths).  Even if he does, the ladder won't stay put as it's the length of the diameter but has width, so can't be erected vertically and due to the frictionless surface, can't wedge the ladder.

Therefore he can't climb the ladder.

I did a simple drawing to illustrate the problem, but couldn't see how to copy it from Flickr into the new format forum !!!! the HTML link seems to have gone AWOL !!

Don,

Quis vexat vexator?

Ladder/sewer,

I guess the theoretical answer is that as he climbed, the ladder would revolve, so that when he got to the "top" of the ladder, he would be back where he started but upside down.

I'm sure I once saw a circus performer climb to the top of an unsupported ladder just by subtle alterations to the angle of the ladder as a counterbalance to his weight. Presumably if the sewer worker used to work in a circus, he could do something similar..........

It reminds me of an old rhyme:-

"A man beside the sewer sat, and beside the sewer he died, and at the coroner's inquest, they called it sewer-cide"

Steve D

Spot-on Steve, well done !

Cheers, Don

Huge posted:

Don,

Quis vexat vexator?

I realised, from your initial response, that I had badly worded the question. To assist others and yourself I felt the easiest way to clarify, was to highlight the two assumptions that you had very reasonably made, (but which consequently were wrong) and clarify my original concept.

Hopefully this worked. But knowing how to link to Flickr in this new forum would also be helpful (for me anyway)

Don Atkinson posted:

Huge posted:

Don,

Quis vexat vexator?

I realised, from your initial response, that I had badly worded the question. To assist others and yourself I felt the easiest way to clarify, was to highlight the two assumptions that you had very reasonably made, (but which consequently were wrong) and clarify my original concept.

Hopefully this worked. But knowing how to link to Flickr in this new forum would also be helpful (for me anyway)

I was using vexo in its primary meaning of tease or provoke, rather than the more aggressive meanings.

So who teases the teaser?

I didn't think you were being aggressive Huge. Ater all, my Latin is still limited to amo, amas amat amamus amatus amant............so I assumed what you were try to say, in a roundabout way, had something to do with love 

The missing picture...............

A couple of postcards early 20th century

Not too good on the old French these days...............

We start this morning, 25 kilometers by foot. I assure you that on arrival at......

johnG posted:

Further to the lorry fuel consumption problem I worked it in Mathematica for 'fun' and for interest show some of the working:

lorrydata references the data in table form.

Fit[lorrydata, {1, x, x^2}, x]     - the Fit function does the least squares stuff and returns the expression:

20.483 - 2.67236 x + 0.114133 x^2

 This expression plotted over the data points to see the good fit.

Then declare a function of the above expression:

f[x_] := 20.48 - (2.67 x) + (0.11 (x^2))

and solve for the load

f[7.6]

answer:  Fuel 6.5416   (km/litre) 

Beats the hell out of excel.

 

Hi John.

Yes, you can fit almost any curve to the data.

Excel will also fit a second order polynomial to the data set and provides the following equation for the curve :-

y (fuel) = 0.1163x² - 2.7186x + 20.702

which translates with x = 7.6 into

y (fuel) = 6.76 km/litre

My linear equation was y (fuel) = 14.451 - 0.97795x

Excel's linear equation was y (fuel) = 14.469 - 0.9808x

I'm still not sure why Excel generates slightly different equations.

Don, don't forget that, unless there's a perfect fit to the line, you'll get slightly different values dependant on whether you do the regression of X on Y or the regression of Y on X.

Huge posted:

Don, don't forget that, unless there's a perfect fit to the line, you'll get slightly different values dependant on whether you do the regression of X on Y or the regression of Y on X.

Huge, you beat me to it. That was going to be my next "Teaser"

ie same data set, but with the fuel consumption as the Independent variable and the load as the Dependent variable !

Don Atkinson posted:

Huge posted:

Don, don't forget that, unless there's a perfect fit to the line, you'll get slightly different values dependant on whether you do the regression of X on Y or the regression of Y on X.

Huge, you beat me to it. That was going to be my next "Teaser"

ie same data set, but with the fuel consumption as the Independent variable and the load as the Dependent variable !

 

Linear regression by least squares is deterministic, so if the data*, the function to be fitted and the calculation precision are the same, you should get the same results for the parameters.  However if the precision is different (e.g. 32bit float against 64bit float), that might possibly account for the relatively minor differences.

* Reversing the dependent and independent datasets is, technically, a change to the data - hence different results!

On the other hand non-linear regression is non-deterministic; and despite the sheer brilliance of the Levenberg-Marquardt algorithm; it can't fix fundamental uncertainties arising from chaos theory, no matter what termination conditions are used to end the iteration.

Huge posted:

Don Atkinson posted:

Huge posted:

Don, don't forget that, unless there's a perfect fit to the line, you'll get slightly different values dependant on whether you do the regression of X on Y or the regression of Y on X.

Huge, you beat me to it. That was going to be my next "Teaser"

ie same data set, but with the fuel consumption as the Independent variable and the load as the Dependent variable !

 

Linear regression by least squares is deterministic, so if the data*, the function to be fitted and the calculation precision are the same, you should get the same results for the parameters.  However if the precision is different (e.g. 32bit float against 64bit float), that might possibly account for the relatively minor differences.

* Reversing the dependent and independent datasets is, technically, a change to the data - hence different results!

On the other hand non-linear regression is non-deterministic; and despite the sheer brilliance of the Levenberg-Marquardt algorithm; it can't fix fundamental uncertainties arising from chaos theory, no matter what termination conditions are used to end the iteration.

Hi Huge,

I think saying "well done" would be a bit patronising in your case, so I won't, but thank you so much for sharing your insight above.

When I set the initial Load v Fuel question it was, like the other questions in this thread, meant to take us (well, some of us anyway ) back to our school days.

My immediate follow-up to the initial question was intended to draw out the common (?) error of then using the regression in the reverse direction. As you say, Load v Fuel is a different data set to Fuel v Load.

The principal difference between my "hand" calculations and the Excel calculations is the reversal of data. ie all the answers I gave for Excel were based on the Fuel v Load data ie Fuel was the independent variable. There might well be other differences (as you rightly suggest, as well) but I can't tell because I don't have two computers working at different precisions.

John's idea to fit a second order polynomial curve gives the same answer as Excel, again to about 5 sig figs - assuming we use the same data set and not a reversal.

I'm pleased somebody identified the regression of X on Y is different to  the regression of Y on X. Makes it so much more convincing.

Hi Tony,

I downloaded the Gridshading puzzle and have now created the QR Code.

I don't have a QR Reader and my mobile phone is a Nokia similar to the one Noah took on board the Ark. and I can't see an "app" that runs on Windows XP or Windows 95

Look like I have to invest £500 in an iPhone to be considered for that new top-job at the end of January

Back to GCSE, it was GCE “O” Level when I was at school. So, not too difficult. Well Exams have been getting easier for decades, haven’t they !............or have they

Recurring decimals. You know the type 1.666666……representing 1 and 2/3

Of course, sometimes it is a string of numbers that form the recurrence eg 0.285714285714 and that are repeated ad-infinitum.

Many of us will recognise at a glance the fraction that 0.285714...... represents. But how about 0.234234234.........?