Mad as a Hatter
100 mathematicians are standing in a line, wearing a black or white hat. Each mathematician can ONLY see the color of the hats of the people in front of them. So the first person sees no hats, the last sees 99.
The mathematicians are allowed to talk to each other and decide upon a strategy, for a government rep is coming to cut off funding. Each person can only say “black” or “white.” If you correctly say what color hat you’re wearing, your funding is continued and you live. If you’re wrong, you lose your funding, and you may as well be dead.
How many mathematicians can you guarantee will keep their funding?
You are not allowed you use “tricks,” say a person delays one second before answering means A, two seconds means B, … You have to answer IMMEDIATELY what color hat you’re wearing.
Communicated by Gergely Harcos.
Your email doesn’t work. email me at [email protected]
You can do better….
50% would get it right… right??
david: correct — email me at sjm1 AT williams.edu
John: close to optimal but one can do a bit better (feel free to email me at sjm1 AT williams.edu, as I don’t want to post solns as that can spoil other people’s fun)
John: correct! Not posting as it’s the soln (email me at sjm1 AT williams.edu to chat more)
ahh, but there is a better way wi’out cheating…. sjm1 AT williams.edu
I take of my hat and take a look at it, therefore every mathematician will keep its funding
Alex Irby: One can do better. If you want to chat about problems please email me at sjm1 AT williams.edu
two problems: (1) you don’t know there are the same number of black and white hats, and (2) they cannot move. //s
PS: email to you bounces, email me at sjm1 AT williams.edu
two problems: (1) you don’t know there are the same number of black and white hats, and (2) they cannot move. //s
If they had time could they rearrange themselves alternating between black and white, then they could see the person in front of them and say the opposite color of the person in front of themselves hat, then if there was not an even amount of people all the extra people could go to the back of the line after asking what their color was. Assuming everyone was honest and no one had a very short memory 100% of the people would be correct. I think…:)
andreas: correct, not posting as it’s the right answer. email me at sjm1 AT williams.edu if you want an interesting answer that’s almost as good as what you have
let me try sending you a hint first. //s
May I please have the solution to Mad as a Hatter? Thanks.
ah, but their honor does….
i’m sorry but nowhere does it say that the mathamations can’t just tell each other ” hey bill you have a black hat!” “thanks Rodger you too have a black hat” and thus everyone keeps their funding.
I can’t follow your logic — can you be more explicit? Send me an email at [email protected]. For example, what does ‘person behind him’ mean? There is no one behind the last person….
99 can keep their funding as person behind him can see his hat color only 100th person is in delimma 🙁
Carl — well done! Not posting your response as it’s the answer. /s
do end to beginning — last person who sees all speaks first.
what is the order in which they are asked the color of their hat? is it from beginning to end, end to beginning, random? it makes a difference
NOAH: great job. You can do better than what you wrote, but that is VERY close to optimal. Email me at [email protected] if you want to discuss the riddles more.
yes! can you do better! you had the yes/no right as well
lets say person100 will say the hat color of person 99
person99 will say the hat color person100 say
person98 will say the hat color of person 97 AND SO ON
that way more than 50% will be saved from death
Not quite following what you’ve written; also, the soln has to work for ANY number of white and black hats. hint: can you find a way to always ensure at least 50%? At least 66%?
i know how to solve this riddle with 50 black and 50 white hats. using this method would give you a chance of 100-(difference between black and white hats) to get the answer right, depending on where in line you are.
Aldo: correct, well done
Nice generalization — thoughts all?
To make this problem a little more interesting, can I tweak it a little bit?
Say, same rule and everything, but now there arent only black or white hats. Say there are 100 colors of hats to choose from. Colors of hat are determined randomly to assign to each person. No minimum hats of a certain color. So say purple is one of the 100 colors, but it might be by chance that no one was assigned purple.
Have fun! 😀
You don’t get to decide on the amount of black or white hats — the sadistic (is it redundant to say sadistic?) professor does that!
If you email me which mediums you’re most interested in I’ll send along the solutions. /s
Can we decide on the amount of white and black hats? if so, I can get 99 out of the 100 to keep their funding.
Also, I have been looking for a site like this with real math (non-word trick) riddles. And you are correct on the front page that I would not spend as much time on the riddles if I had the answer right away. However, I have looked over all the mediums, would it be ok if you sent me most of the mediums’ answers if not all of them?
Keep it up. I would love to see more.
Usually it’s to make it dramatic and have the stakes high so people take these riddles seriously. One can, of course, rephrase.
Why does almost every game have to do with killing? Ponies!!!!!!!!!!