Earlier today I set you these two questions, in the spirit of Lewis Carroll, who died 125 years ago this week.
1. A handy chat
Cheshire Cat: Okay Alice, let us call a whole number ‘handy’ if its digits add to a multiple of 5. Can you think of a handy number?
Alice: Why I can think of plenty! 14, 55, 406, 77777…
CC: Very good! But your examples are all quite far apart. How close together do you think two handy numbers can be?
Alice: Well… 55 and 64 are both handy, and they’re pretty close. Will that do?
CC: I think you can do better than that.
Alice: Let me think…
What’s the best you can do?
Alice: I’ve got it! I found a pair of handy numbers right next to each other!
CC: What, just 1 apart?! Prove it – what are the numbers?
Alice: I found lots of pairs, but I think 49999 and 50000 is my favourite.
CC: Bravo, Alice, bravo! I shall have to return with something a little more challenging…
2. A cardy convo
Four cards numbered 1, 10, 100 and 1000 lie face down on a table. The cards are distributed between three truth-tellers and one liar, with each person receiving one card. Then the four of them speak one after another:
“My number is odd.”
“My number has 3 digits.”
“My number is less than 100.”
“My number is more than 100.”
What number was on the liar’s card?
Solution The liar has the 10 card
If the liar has the 1, then neither the liar nor the truth-tellers can say “My number is odd”.
If the liar has the 100, then neither the liar nor the truth-tellers can say “My number has 3 digits”.
If the liar has the 1000, then neither the liar nor the truth-tellers can say “My number is more than 100”.
So the liar must have the 10. This is how the conversation must have gone:
Truth-teller – 1 – “My number is less than 100”
Liar – 10 – “My number is odd”
Truth-teller – 100 – “My number has 3 digits”
Truth-teller – 1000 – “My number is more than 100”
Thanks again to Daniel Griller. His latest puzzle book, A Ring of Cats and Dogs, is out now.
I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.
I give school talks about maths and puzzles (online and in person). If your school is interested please get in touch.