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A (temporary) break from dominoes and wheels …

We Hiked up a Mountain.

Myself and Mrs D hiked up a mountain whilst we were in the Rockies this summer. Well, we hiked up several, but we’ll just stick with this specific one !

Going uphill we managed a rock steady one and a half miles per hour. Coming back down we achieved a staggering four and a half miles per hour. The round trip took us six hours.

How far was it from the start to the top of the mountain ? (or from the top to the bottom !)

1 mile?

Hi Mike, your answer suggests six hours to cover two miles (one mile there and another mile back )?

You’re going to have to give this one another look Mike - sorry.

PS. just to avoid any angst, there are no silly tricks. The walk could well have been along a dead level canal, but I would then have to invent a different story to justify the speed difference outbound and inbound !

Ah yes. 3 times faster going downhill. 1.5 hrs down, 4.5 hrs going up, I had it the wrong way round. So, 1.5 hrs down at 4.5 miles/hr = 6.75 miles?

You can now relax and enjoy Sunday at leisure …
… playing with that new set of dominoes…:sunglasses:

Well done !

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For the seventh domino (2-2 given plus 5 already identified above) I could see two possible locations for “0-0”. these are shown in Green and Blue below.

If I selected the Blue one, it would deny me using either of the 4-0 or 0-4 options associated with those Zeros. I couldn’t see any other 0-4 or 4-0 pairs available.

I therefore selected the 0-0 domino shown in Green below.

Now I had to choose between upper 4-0 or the lower 0-4 just the the right of the given 2-2.

Whichever option I chose, would mean the other option must NOT form a domino (you can’t have two 4-0 dominoes in a set) and as a consequence, would force the pairings of quite few dominoes in that “eastern” arm of the layout.

I opted for the …

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Sorry, I haven’t done this yet, but had concluded you would have to find the option with the least scenarios and then trail and error to rule one out.

Hi Mike,

I’ve only outlined the easy part above. But the more difficult part isn’t really all that hard to sort out.

…I opted for the…

… the upper 4-0 as shown above in Blue.

The next half dozen dominoes then fell into place, mainly on the basis of preventing “orphan” tiles being left around.

Having selected the upper version of domino 4-0, the 3-1 followed. then the 0-3 (otherwise there would be two 4-0’s !) The 5-2 and 4-6 were also forced to avoid orphans and doing so, eliminated a possible 2-3.

The 2-3 (above the given 2-2) was next simply because the only other 2-3 had just been eliminated. The 5-6 followed to avoid orphans !

More or less half way there now !

Of the tiles that were left, I could only find one pair that would make a unique domino. All the other tiles could make at least two versions of the outstanding dominoes.

The unique pair of tiles made the 2-4 domino, shown in Blue.

This allowed me to subsequently pair up tiles to form another half dozen dominoes.

Divide 45 …

Divide 45 into four parts, such that the first part plus two, the second part minus two, the third part multiplied by two, and the fourth part divided by two all equal each other.

8,12,5,20
I realised that I needed a root number which all the sums equalled, call it x and the 4 parts a,b,c,d
A+2=x
B-2=x
C2=x
D/2=x
Thus, A+B=2
x, C=x/2, D=2x.
A+B+C+D=45, and also from above = 2
x +x/2 + 2*x = 4.5x.
Therefore X=10.

Nicely done Eoink.

Your method is much neater than mine.

I will post mine later, but it’s a bit clumsy.

My Solution (which delivers the same outcome as Eoink’s solution)

Given :-

a + b + c + d = 45 Eq (1)

a + 2 = b – 2 = 2c = d/2 Eq (2)

Hence b, c, d in terms of a :-

b = a + 4 Eq (3)

c = (a = 2)/2 Eq (4)

d = 2(a + 2) Eq (5)

Substituting (3), (4), (5) in (1) :-

a + (a + 4) + (a + 2)/2 + 2(a + 2) = 45

4.5a + 9 = 45 or 4.5a = 36

a = 8

Therefore substituting for a = 8 in (3), (4), (5) :-

b = 12
c = 5
d = 20

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Oooopps

That line that says c = (a = 2)/2 Eq(4)

should read c = (a + 2)/2 Eq(4)

“+” and “=” are on the same key !!!

Here is my original hand-written solution.

I slightly modified the workings when typing out above, hoping to make things clearer.

No need to feel the need to type/powerpoint/Word/Excel any solutions, just scan or photograph the manuscript !

Well, here are the next half dozen dominoes that I identified.

I started with the 3-4 domino and worked clockwise around the 2-4 domino ending with the 3-5

A few wines on a Saturday night and I think I cracked the dominos…

Well Mike, it’s still Saturday morning here in the UK and the weather is low cloud and drizzle, so we’re sitting on the ground !

It sure looks as if you’ve cracked this one so another couple of wines to celebrate and relax.

Actually, I find that most of the Brain Teasers are in themselves, quite relaxing and go quite well with a glass of nice wine.

As kenc would say…
…enjoy !
and well done !!

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