Brain Teasers are Back!

Well done Mike, well done Dozey.

Yes, the 9th power of any number always ends with the same digit as the initial number.

The same is true for the 5th power and the 13th power and 17th power…ie every 4th increase in power starting with “1”

There are repetitive endings for most (if not all) powers eg if the initial number “n” ends in 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 then the last digit of n^2 will be 0, 1, 4, 9, 6, 5, 6, 9,4 ,1 etc and the last digits of n^(2+4) will also end in 0, 1, 4, 9, 6, 5, 6, 9,4 ,1

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Anybody tried the “Frogs and Toads” teaser above ?

I first encountered this problem 25 years ago when my eldest brought it home as a “family” project for the week ! She was about 15 at the time.

It took about three days of persuading before we got her to write the moves down and count them !

By the end of the week we had developed the project quite a bit further…:sunglasses:

The Frogs and Toads can change places in 15 individual moves, assuming there are three of each.

What about one of each, or two of each or perhaps four of each ?

Not sure if the Frogs & Toads teaser is too easy, too difficult or too badly written ?

One a side takes three moves, two each side takes eight moves. Three a side takes fifteen and four each side requires twenty four moves.

There is a neat formula that derives these figures.

Anybody ?

Perhaps 5 will need 35 moves?

Formula is x(x+2) where x is the number on each side

Actually the amount of digits in the number is irrelevant, if they add up to 9 then that number is a multiple of 9.

I thought that was the case, but didn’t take the time to check. I find it very interesting. Perhaps it is to do with our number system being base 10?

Perhaps you’re right Dozey…

…well done !

Hi Steve,

Good formula !

That formula can be modified to suit a variation where the number of Frogs and the number of Toads are unequal. Eg 2 Frogs and 3 Toads.

Just a guess based on the first formula, with no mathematical thought or proof:-
xy+x+y (where x = Frogs and y = Toads)

You got it Steve. Well done !

I usually write it down as f + ft + t where f and t represent Frogs and Toads respectively.

But apart from checking the formula against half a dozen Frogs and Toads and all combinations in between, I also don’t have any real proof.

Now that Parliament has been suspended, I thought it might make a nice break to put politics to one side again, and give the old brains a rest for five weeks … :sunglasses:

The diagram below shows a standard set of 28 dominoes. They have been set out in a neat pattern.

I have drawn in the edges of one domino, ie 6-4.

All you have to do is draw in the edges of the other 27 !

Rather than hand copying the diagram, it should be possible to copy and paste into PowerPoint or Word.

I also created a list of standard dominoes so as to tick them off as I progressed.

I’d give it a go, but I have no idea how dominos work…

Mike ! You cannot be serious (as McEnroe once said !)

If you are serious, I will try to explain, and I will scan a copy of my list of the 28 dominoes that form the standard set. 6-6; 6-5; 6-4: … 5-5: 5-4: 5-3: … 1-1: 1-0: 0-0.

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Great, on to it.

Can the darn things be reversed? There is only one position for 6-6, but 6-2 needs to be reversed?

Yes, the “numbers” can be reversed. The real dominoes don’t have actual “numbers”, they have “spots” that correspond to the numbers.

This means that real dominoes can be looked at L-R or R-L. So 6-2 and 2-6 would be the same domino.

Hope this helps ?