Brain Teasers are Back!

I like puzzles that seemingly have very little info to work from, such as ABCDE/EDCBA = 4 being a single equation with five unknowns.

My solution starts off in the same way as the 4-digit teaser. That is, A=8 and E=2 for the same reasons as before.

We now have:
80000 + 1000B + 100C + 10D + 2 = 4x20000 + 4000D + 400C + 40B + 32

It simplifies to:
96B - 30C - 3 = 399D, so D must be odd.

Max value of left hand side is 96x9 - 30x1 - 3 = 831, so D = 1 (as 3x399 would be too high)

We now have:
48B - 15C = 201, so C must be odd (3, 5, 7 or 9)

Trial and error (and 4th attempt) shows that C = 9 is the only one that gives an integer for B:
48x7 - 15x9 = 201

Answer is 87912/21978 = 4

And I didn’t need to use Douglas Adams’ this time to spot the answer!

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Well done Ravvie.

Very nicely explained.

And this time, all without the need for Douglas Adams’ !

I assume we all know the probability of winning the UK National Lottery ?

When it started, way back in 1994 (?) you had to pick any six different numbers between 1 and 49 inclusive. If those six numbers (regardless of sequence) matched the six numbers selected at random by the machine, you were a winner. The probability of winning was … well, you probably know … (?)

A few years ago the rules changed. You could now choose your six numbers from a wider selection of numbers viz 1 to 59 inclusive. Your chances of winning the jackpot therefore “increased” or “decreased” compared to the initial game ? and now stood at … well again, you probably know (?).

One mark for looking the answers up.
Two marks for having remembered them (no cheating !!)
Ten marks for providing the necessary arithmetic.

The first number is any of 59 different numbers, the second number is any of 58 numbers, etc., so 1/59 x 1/58 x 1/57…x 1/54. As the 6 numbers are picked irrespective of order, you multiply that by 6!, or 720.

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Very neatly explained JR. Full ten marks :sunglasses:

So that anyone trying the arithmetic can feel confident, the answer is approximately 1 in 45 million.

And for the the original Lottery, the probability of the big win was approximately 1 in 14 million.

Simplified Orbits. (or going around in ever decreasing circles ?)

2 Earth years I think. For each Earth year, Mercury and Earth are aligned with the original starting alignment. However Venus is on 1.5 cycle afters after an Earth year, and takes another Earth year to be on 3 cycles, in an aligned position.

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Well done Mike. Spot on and an easy to follow explanation.

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My eldest grandson ran in an inter- school x-country race this morning.

If one other runner had finished ahead of him, there would have been as many runners behind as finished in front of him.

If one more runner had finished behind him, there would have been three times as many finishing behind him, as finished in front.

How many runners took part in the race, and in what place did my grandson finish ?

There were 9 people in the race and your grandson came 4th.
I guess it must be possible to solve it intuitively, but I resorted to simultaneous equations.
Many thanks for all your teasers this year Don.

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Well done Steve. Especially generating simultaneous equations !
I’m afraid that on this one, I resorted to intuition and progressive trial and error.

The second paragraph shows there has to be an odd number of runners (equal numbers behind and in front, plus grandson).

I then started with the possibility of 3 runners, and progressed through 5, 7, 9 and “bingo”

But your method is far more rigorous. Well done.

Thank you Steve. It’s been a real pleasure.

I hope all the contributors to the thread have enjoyed at least one or two of my teasers. I have certainly enjoyed the teasers posted by yourself and others.

And likewise, I hope that yet others, who may not have posted but might have been watching or playing along from the side-lines, have also enjoyed the thread.

Cheers
Don

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GCHQ Christmas Challenge.

Me and my daughter managed 6 between us. But, I must admit I only managed to contribute one answer. :grin:

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Thank you for reminding us all about this year’s set of challenging puzzles from GCHQ

It took me quite an effort to sort out number 4 and number 6. The others fell into place fairly straight forward.

Definitely enjoyable and well worth having a go even if you don’t intend to apply for a job :man_detective:

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A couple of (relatively) easy teasers (compared to FatCat’s offering !)

Pyramid Numbers.

The number in each circle is the sum of the numbers in the two circles below.

Simply fill in the missing numbers.

Simple Sums

WARNING !!

Do NOT give this “Teaser” to your school-children or grand-children !!

The instructions (below) make it clear that for this Teaser, you MUST do the calculations in the sequence they appear. So, the top line of the teaser reads: 5 + 9 = 14: 14 x 3 = 42: 42 – 6 = 36

This is NOT the standard form of doing arithmetic where the sequence is always :

Brackets/Indices/Division/Multiplication/Add/Subtract.

Using conventional arithmetic, the top line would normally be evaluated :

5 + (9 x 3) – 6 = 5 + 27 – 6 = 26

The teaser might confuse us, but please don’t let it confuse your school children.

Arrange one of each of the following four numbers (3, 5, 6, 9) and one each of these symbols X (multiply); + (add); - (subtract) in every row and every column so as to achieve the answer at the end of each column and row. Do the calculations in the sequence in which they appear. (Do not use the standard form of doing arithmetic)

Many thanks to Mrs R for both the maths and the technology. I hope we understood the rules correctly.

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Well done Ravvie, or rather Mrs R !!

Looks like you both fully understood the rules. Hopefully the WARNING that I posted didn’t appear
condescending. I have avoided using this sort of puzzle with my grandchildren for reasons that both yourself and no doubt Mrs R appreciate.

Cheers
Don

Simple Sums

WARNING !!

Do NOT give this “Teaser” to your school-children or grand-children !!

The instructions (below) make it clear that for this Teaser, you MUST do the calculations in the sequence they appear. So, the top line of the teaser reads: 6 x 9 = 54; - 4 = 50; + 7 = 57

This is NOT the standard form of doing arithmetic where the sequence is always :

Brackets/Indices/Division/Multiplication/Add/Subtract.

However, using conventional arithmetic in this particular case, the top line would also be evaluated the same way. But, the rest of the matrix MUST be evaluated in the sequence in which the numbers appear. A few lines might give the same answer either way, but don’t rely on it !

The teaser might confuse us, but please don’t let it confuse your school children.

Arrange one of each of the following four numbers (4, 6, 7, 9) and one each of these symbols X (multiply); + (add); - (subtract) in every row and every column so as to achieve the answer at the end of each column and row. Do the calculations in the sequence in which they appear. (Do not use the standard form of doing arithmetic)

We were bagging up a pile of 2p pieces today with the help of our young grandsons. They inevitably started playing and came up with the triangular patterns below.

Our older grandson, happened to mention he was practising sequences in maths at school. Why he picked ‘58’ is anyone’s guess, but I asked him to tell us the answer, and have retained the question as was !

PS. I gave him a pencil and some paper, and he came up with the correct answer in less than two minutes :sunglasses: