Winner of Best Visual Illusion 2010

Truly bizarre…

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Grass Track Illusion

Believe it or not, these two photographs are identical….

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Puzzle 18 – Solution

The Numbers Game

The optimum solution is to choose a number between 1 and 5 based on the following probabilities :

p(1) = 1/16
p(2) = 5/16
p(3) = 4/16
p(4) = 5/16
p(5) = 1/16
p(>5) = 0

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Puzzle 18 – Numbers Game

Andy and Bob are playing a game. Each person writes a positive integer on a slip of paper. The person who wrote the smaller number wins 1 point, unless it is smaller by one, in which case the person who wrote the larger number wins 2 points. The game is fair in the sense that neither player has an advantage, but what strategy is best in the sense that no other strategy can beat it in the long run, and if any different strategy is followed, there is a counter-strategy that can beat it?

Solution will be on Sunday.

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Weekly puzzle 17 – solution

Gunfight

Part A

Incredibly, Charlie should just shoot in the air!

Part B

Despite being the weakest shot, Charlie has the best chance to survive.

Part C

Andy : 3/10

Bob : 8/45

Charlie : 47/90

Part D

If Charlie’s probability is above 80%, Bob gets the best chance of survival since Andy will now aim at Charlie. If Charlie’s probability falls below 36.18%, Andy has the best chance of survival.

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Weekly puzzle 17

Gunfight

Andy, Bob and Charlie agree to a gunfight under the following conditions. After drawing lots to determine who fires first, second and third, they stand on an equilateral triangle. It is agreed they will fire single shots in turn and continue in the same order until only one survives. At each turn, the man who is firing may take aim wherever he pleases. All three duelists know that Andy always hits his target, Bob is 80pc, and Charlie is the weakest shooter at 50pc. All three are infinitely clever.

Part A

What should Charlie do when his turn comes around ?

Part B

Who has the best survival chance ?

Part C

What are the exact survival probabilities for each man.

Part D

Is it possible to change Charlie’s accuracy in order to give a different man (from Part B) the best survival chance ? If so, how? If not, why?

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Optical Illusion Thatcherism

The following extract has been taken from BBC Bang

Most people can still recognise a face that’s been rotated upside-down, and they still do even when altered in rather major ways. In particular, a distortion known as ‘Thatcherization’ often goes unnoticed – that’s where just the eyes and the mouth are changed to be the right way up.

If the entire face is then rotated into the normal position, you’re in for a shock as the face suddenly looks grotesque. Try rotating the line-up of suspects below – I’m particularly happy with how awful my own Thatcherized face suddenly becomes!

With photo manipulation programs, you can easily do the same to your own digital photos, but I’ve found some faces and expressions aren’t so good.

Below are the other presenters, along with a face that definitely doesn’t Thatcherize well. I suspect this is because the smile takes up a good portion of the lower part of the face, approaching the edges of her cheeks.

This problem (if having a great smile is a problem!) also seems to affect Liz’s picture, but to a lesser extent.

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Weekly puzzle 16 – solution

Free Gas!

The correct answer is 9 miles from Andy’s house.

Each round trip is 21 miles. Clearly some common sense assumptions need to be made about the size of a gas tank. For example, if he makes 15,873 round trips before filling up with gas, that is a total of 15873*21=333333 miles. He gets 15 miles per gallon, so he used 333333/15=22222.2 gallons. Gas cost $.30/gallon, so the bill comes to 0.3*22222.2=$6666.66

Clearly this is not practical, and the cost of purchasing a vehicle with 22000 gallon tank would far out-way the free gas. And I don’t think the gas station manager would be too happy when that huge thing rolled up. And as for why Andy wants to commute in a 22000 gallon monster I don’t know. Also, the time it takes him to fill it up would mean he either has to get up very early one morning, be late for work, or have his wife put his dinner in the microwave. But since it would only happen once every 60 years or so, he would be retiring before the first tank load ran out ….

So, we can safely assume that the maximum gas is $9.99 which would equate to a tank of 9.99/30 = 33.3 gallons.

This means that the price (and the number of miles between fillings) is a multiple of 37.

Now, if Andy were to fill his tank always on the same leg, he would need to travel an number of miles that would be a multiple of 21 and 37, i.e. at least 777, which gives a price of $15.54 and a tank capacity of 51.8 gallons, which is too large.

We can therefore assume that Andy fills his tank alternately on the outbound and on the inbound legs. Let us assume that the station is a miles from one end of the road and b = 10.5 – a miles from the other end. The distances between fillings are p*21+2a and q*21+2b.

After two fillings the total distance travelled is (p+q+1)*21, and this must be a multiple of 37, which implies that p+q+1 is a multiple of 37.

Because we want the number of miles between fillings to be less than or equal to 444, p and q must both be <= 21. Therefore we must have p+q = 36, and p and q must be between 15 and 21.

As p*21+2a must be a multiple of 111 and 2a <= 21, the only possible solution (excluding symmetry) is p = 21, 2a = 3, q = 15, 2b = 18. the distances are 444 and 333 miles, and the prices are $8.88 and $6.66.

As the station is an integral number of miles from Andy's home, the distance in question is b = 9 miles.

Note that the tank capacity is still 29.6 gallons, which is still quite large. Therefore we have established that Andy is a Texan.

Extra Note : Some mathematicians have argued that the puzzle was not explicitly defined, and that there are infinitely many solutions i.e. multiples of the 22,000 example above. However, it should noted that every reader made correct assumptions about the gas tank to solve the puzzle, and mentioned the size as a humourous addition. This is a puzzle site, not a Math site.

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Candlelight illusion

Similar to the previous post but more beautiful…

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Weekly puzzle 16

Free Gas!

Andy drives to work, and then home the same way, for a total round trip of 21 miles every day. Along the route is a gas station which is offering free gas if, when filling up a tank to the very top, the price consists of repetitive digits, i.e. $1.11, $2.22 etc..

Gas costs 30 cents per gallon and Andy knows that his car does exactly 15 miles to the gallon. Assuming Andy does no other driving and takes the same exact route every day, he works out that once he fills his tank at the station, he can get all subsequent gas free.

If the station is an integral number of miles from Andy’s home, how far away is it?

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Optical illusion in house

What I want to know is how long this took to create!?

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Weekly puzzle 15 – solution

Drunk

The man got drunk at lunch time so it is still light.

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More wonderful Ricky Jay…

After a serious but breathtaking card display that I posted a week ago, here is Ricky Jay at his entertaining best, taken from his NBC special. No comments on the YouTube title, please!

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Startling illusion

Found this on a blog I follow, truly remarkable. The image you see below is NOT a spiral !

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Weekly puzzle 15

Lateral this week, this one you will either get straight away or be pondering for hours…

Drunk

A man is walking home drunk from the pub. He is wearing black trousers, and a black coat. He is so drunk that he wanders in the road and is staggering drunkenly down the centre of it. A car comes towards him at high speed but manages to avoid him. How ?

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