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

My initial thoughts (outlined above) lead to 450

Any better offers ?

Been lurking for a while but thought it was time I joined the discussion.

All depends on how you read the question...

With digits 0-9 (10 digits) and one digit used once, the other 4 times, I agree with Don at 450 combinations.

Not remembering whether it was 1,2,3 or 4 of whichever digits it would be 1350 combinations.

But if we are down to nine digits...

I arrived at my 450 as follows :-

Selecting two different numbers from 0 to 9 can be done in 10*9=90 ways.

Arranging those numbers can be done in 5 ways vis • − − − −      − • − − −   − − • − −   − − − • −   − − − − •

Hence 90*5=450 possible combinations for Tony to try.

There is another way of arriving at the same solution, but the one above seems more easy to grasp.

Hi Matthew, was your logic similar to mine ?

Don

I don't think Tony's brain teaser is a mathematical puzzle.

Tony, only the numbers 0,1,2,3,4,5,6,7,8 or 9 could be selected at random for the "first" digit ie a choice of 10.

The "second" digit can then only be selected at random from the remaining 9 digits.

Hence the choice of two digits can only be made in 90 possible ways. Selecting 6,1 is a different choice than 1,6.

There is nothing "random" about the arithmetic, only the choice of digits.

A bit like selecting two cards from a standard deck. ie 52x51 possible selections (without replacement) and assuming that the order eg Ace of Hearts; 5 of Clubs is different to 5 of Clubs; Ace of Hearts.

I know

But one or two on-lookers might be interested (or ammused if we got it wrong) in how Matthew (probably) and myself came up with 450.

As a member of this fine forum Tony surely had the ability to hear and remember the five tones. So he knows which one of them is the different one. Now he has to press up to nine of the ten numbers to find the tones hehas heard. If he recognizes only one, the second one belongs to the single number he has not pressed. This adds up to 9 + 5 = 14.

If the sedatives interfered with his otherwise impeccable hearing, I agree with Don. For each of the five possible places the odd number can take, there is a choice between 10 numbers for the odd number, followed by 9 possibilities for the other four numbers. That would be 5 * 10 * 9 = 450

Mulberry,

I see where you are coming from and I like it !  However............

With Tony's super-hearing, he already knows the sequence in which the two tones he has heard were pressed. So he only needs to use his super-douper-hearing to enable him to detect which two tones were used. As you rightly say, he only needs to hear 9 of the tones (at most) in order to deduce which two these were.

So with his Souper-Douper-Hearing Tony only need to press, at the most, 9 buttons.

However, with his X-Ray eyes , he has already detected that the two numbers being pressed were 6 and 1........... and he also noticed that the 6 was only pressed once whilst the 1 was pressed four times. So I guess that reduces Tony's need to only press one sequence of buttons to gain entry. Why he didn't mention this in his opening post I have no idea !

For the rest of us mere mortals, devoid of golden ears and x-ray eyes, I'm afraid its the Full Monty ie 450.

Nice one Tony ! How's the toothache ?

Crossed posts. I started mine just before the 10 O'clock News and finished it shortly after, so hadn't seen Tony's interleaving response.

Cheers

Don

He he he! Never laughed so much... and tooth is fine after a course of Amoxicillin!

Cheers,

Tony

Hi Don and Tony,

My logic followed that of Don's.

Cheers

Matthew

My father was a haulage contractor, so until I left home for university I spent weekends and evenings helping him. Fuel was always a significant cost and fuel consumption varied with a number of factors, including the load on a lorry.

So, if the load on the lorry was 7.6 tonnes, what would be a reasonably reliable estimate for the fuel consumption ?

To help, I have a table of figures based on ten trips.

Load (tonnes)   5.1   5.6   5.9    6.3   6.8   7.4   7.8   8.5   9.1   9.8    

Fuel (km/litre)   9.6   9 .5   8.6    8.0   7.8   6.8   6.7   6.0   5.4   5.4

(Full marks for a Least Squares Linear Regression solution ) - don't it put the clock back

I did the linear regression by hand and then again using Excel (to plot an XY Scatter Diagram and impose a linear trend line)

The equations for the trend line differed slightly as did the evaluation for fuel consumption.

By hand I got 7.1 km per litre (to 2 sig fig)

Using Excel I got 7.0 km per litre (to 2 sig fig)

Not sure yet why there should be such an error when using Excel.............

Hi Don,

after eyballing the latest teaser, your solution seems somehow odd to me. The lorry gets 6.8 km/litre for a load of 7.4 tons and 6.7 km/l for a load of 7.8 tons. How can a load of 7.6 tons lead to 7 or 7.1 km/l? Something like 6.7x seems like the obvious answer here.

I have been wrong before, but still...

Regarding my thoughts on Tonys teaser, I was thinking a maximum of 9 buttons to get the tones, followed by the 5 buttons to open the door = 14 buttons pressed. And yes, I had a good time writing that one

Hi Mulberry,

Re Tony's teaser............

I think Tony was after the "450 answer" ie how many permutations can you form from 10 digits, using one digit once and the other digit 4 times. The order of the digits is obviously important.

The bits about Tony's hearing and eyesight were amusing asides, which I enjoyed. However...............

Tony only asked how many button presses would he have to try, in order to "break the code" and opposed to "gain entry" So i'm sticking with my "9"........................but agree it could take up to 14 to gain entry

Ah ha !

The thing about statistics is that experimental results tend to be a bit variable or "scattered". However, if you draw a graph and plot the results on the graph, you can often see a "Trend". Not always, but often !

The trend might be a straight line or some sort of curved line. You can sketch a line by hand/eye, either straight or curved so that it gives a "best fit" between the plotted experimental results. This "Trend"line can then be used as a predictor of future probable outcomes.

The less scattered the initial results, the more likely the "Trend" line will be a useful predictor of future results.

Give it a try. Just plot the experimental results on some graph paper(*) and see if you can draw a straight line through the results such that on aggregate, the offsets to one side of the line are balance by the offsets to the other side of the line. Then draw the perpendicular from 7.6 tonnes to intercept the line, and read across to the fuel consumption. It should be about 7.0 to 7.1 liters per Km. (*) plot the Tonnes along the "X" axis and the Fuel up the "Y" axis.

Excel can do all this for you if you put the Tonnes and Fuel in two columns and then get it to plot them in an XY Scatter chart and add a "Trend Line". Better still, you can get Excel to plot a "Straight" trend line, or a "Logarithmic" trend line or ..........any one of half a dozen other "curved" trend lines. Some of these curved trend lines predict a fuel consumption figure closer to your 6.7 than the "straight" trend line that I used.

For example

Logarithmic --> 6.87

Polynomial  --> 6.76

Exponential --> 6.87

Statistics and regression only help us make decisions, they don't guarantee results

Global temperatures are rising. We can plot the rise against other events such as the ablation rate of glaciers. Useful ! BUT you can't reliably "Extend" the trend line outside the range of the experimental results. Note the use of the word "reliably" People do, but it all gets a bit "iffy" and sometimes the facts take an unexpected turn which the trend line fails to predict !

Hope all this helps ?

Ooops, hi Mulberry,

On re-reading it, the last post looks like a lecture ! It wasn't meant to be as such. Apologies.

Cheers

Don

Hi Don,

no need to worry. I simply had not enough time to make a graph when I wrote my post. You only saved me from opening Excel myself and doing pretty much what you wrote. But I might do this anyway to get the satisfaction that comes with finding things out .

Hi Mulberry,

Excel isn't exactly intuitive, well not to me ! But its worth persevering - it is a useful programme !

Put the "Fuel" in the left hand column and the "Tonnes" in the right hand column. This way the graph will put Tonnes on the X axis and Fuel on the Y axis which is the way you need it. It shouldn't take too much effort to find the XY Scatter Diagram that Excel offers.

It might take a bit of effort to find the Trend Line options, they drop down from the "Chart" option in the main toolbar.

Cheers

Don

Don Atkinson posted:

Hi Mulberry,

Excel isn't exactly intuitive, well not to me ! But its worth persevering - it is a useful programme !

Put the "Fuel" in the left hand column and the "Tonnes" in the right hand column. This way the graph will put Tonnes on the X axis and Fuel on the Y axis which is the way you need it. It shouldn't take too much effort to find the XY Scatter Diagram that Excel offers.

It might take a bit of effort to find the Trend Line options, they drop down from the "Chart" option in the main toolbar.

Cheers

Don

 

 

You have to love the basic matematical errors in Excel though!

Huge posted:

Don Atkinson posted:

Hi Mulberry,

Excel isn't exactly intuitive, well not to me ! But its worth persevering - it is a useful programme !

Put the "Fuel" in the left hand column and the "Tonnes" in the right hand column. This way the graph will put Tonnes on the X axis and Fuel on the Y axis which is the way you need it. It shouldn't take too much effort to find the XY Scatter Diagram that Excel offers.

It might take a bit of effort to find the Trend Line options, they drop down from the "Chart" option in the main toolbar.

Cheers

Don

 

 

You have to love the basic matematical errors in Excel though!

Absolutely. How their linear regression line is different to mine I just can't imagine. You would have supposed that by now they could get these things right, wouldn't you ?

I mean, they don't employ Hoopla to write their Code, do they ???

Don Atkinson posted:

Huge posted:

Don Atkinson posted:

Hi Mulberry,

Excel isn't exactly intuitive, well not to me ! But its worth persevering - it is a useful programme !

Put the "Fuel" in the left hand column and the "Tonnes" in the right hand column. This way the graph will put Tonnes on the X axis and Fuel on the Y axis which is the way you need it. It shouldn't take too much effort to find the XY Scatter Diagram that Excel offers.

It might take a bit of effort to find the Trend Line options, they drop down from the "Chart" option in the main toolbar.

Cheers

Don

 

 

You have to love the basic matematical errors in Excel though!

Absolutely. How their linear regression line is different to mine I just can't imagine. You would have supposed that by now they could get these things right, wouldn't you ?

I mean, they don't employ Hoopla to write their Code, do they ???

Oh, it's much worse than that...

Column A:  Values 1, <no value>, 3
Column B:  =A1, =A2, =A3

Cell A4: =AVERAGE(A1:A3)
Cell B4: =AVERAGE(B1:B3)

A4 should be the same as B4

I know what you mean, but it does come in handy if you want the average of the numerical entries in Column A, and it saves having to re-enter the data set if you also want the average of all the potential entries, some of which are zero. Perhaps  Quirky?, rather than erroneous, but I agree that it could easily catch out a lot of people !!

Don Atkinson posted:

I know what you mean, but it does come in handy if you want the average of the numerical entries in Column A, and it saves having to re-enter the data set if you also want the average of all the potential entries, some of which are zero. Perhaps  Quirky?, rather than erroneous, but I agree that it could easily catch out a lot of people !!

Hi Don,  I stand by the eroneous.

B2 is assigned the value of A2.
Mathematically, B2 should have the same value as A2, and therefore the averages should be the same.  The error is that B2 does not have the same value as A2.
A2 is 'missing' (as an empty cell it should have the value 'null') but B2 is given the mathematical value zero.  Zero isn't the same value as 'missing' as shown by the averages.