Two puzzles

 AND  NOW  FOR  SOMETHING  COMPLETELY  DIFFERENT

One of my non-writing preoccupations are with mathematical puzzles and I include two here which you will read with considerable interest or immediately ignore depending on your view of such things. The first is not very difficult but the second one certainly is. Both involve the application of pure logic to secure the solution Can you solve them?

1.   Warlocks have captured 4 dwarves and buried them, up to their necks in sand as illustrated above. Three face forward and can not turn their heads, the forth is hidden from them behind a wall. The Warlocks have two red and two blue hats which they place on the 4 dwarves heads in no special order. Each dwarf has no idea of his hats colour. Only one of them is permitted to speak once to say what colour hat he wears, and if he gets it right they all go free. Which one would it be and how would he know 

2.  Ten captives are told by their captors that they will be put to the test tomorrow to see who is to live and who is to die. They will be made to stand in single file all facing forward, and then on each head a hat will be placed which will either be red or blue. The numbers of each colour are unknown. As they are only able to look ahead they will not know their own colour, only those whose backs they face ahead of them. If they can’t guess the correct colour of their own hat they are immediately killed, if they can they are freed .  They have a day to formulate a strategy. They are told interrogation will start with the one at the back who can see all the colours ahead of him, but not his own of course; and work progressively down the line one by one asking each what colour hat he has

First man, in the absence of any information about his hat, has to guess. So he has a 50/50 chance. But his fellows will hear his choice and so he can perhaps give them some coded information from his choice of colour about the hats ahead of him. But each captive may only answer  “red” or “blue” just this one word ,which they can all hear.  There is absolutely no other communication.                                                      How many of them survive with best strategy?

I will reveal both answers later,- but not for a while. 

© John Watts 2018