Matches and rectangles

Before Nov 6 we didn’t have a match stick puzzle for quite a while, so let’s have one again.

1) How many rectangles are there in this square?^**/*****

2) How many rectangles are there in an mxn rectangle?^**/*****
The previous puzzle can be generalized from 3×3 to any size, an m x n rectangle. How many rectangles are there in an mxn rectangle?

New puzzles are published at least once a month on Fridays. Solutions are published after one or more weeks. You are welcome to remark on the difficulty level of the puzzles, discuss alternate solutions, and so on. The difficulty of puzzles is rated on a scale of 1 to 5 stars. You can check your solutions here.

Bongard problem – Dates 2021

In 1967 the Russian scientist M.M. Bongard published a book containing 100 problems. Each problem consists of 12 small boxes: six boxes on the left and six on the right. Each of the six boxes on the left conforms to a certain rule. Each box on the right contradicts this rule. Your task, of course, is to figure out the rule.

1) 2021^**/*****

New puzzles are published at least once a month on Fridays. Solutions are published after one or more weeks. You are welcome to remark on the difficulty level of the puzzles, discuss alternate solutions, and so on. Puzzles are rated on a scale of 1 to 5 stars. You can check your solutions here.

2021

Happy 2021!
May G’d bless you.

I sincerely hope that 2021 will bring an end to the Covid-19 crisis which plagued us in 2020.

For the past couple of year I present you some puzzles which have to do with the number of the new year, in this case 2021.

1) Sum 5^*/*****
how many 4-digit year numbers are there with the sum of the digits is 5? (No leading zero’s allowed)

2) two consecutive primes (i)^**/*****
2021=43*47, two consecutive prime numbers.
There are just two 4-digit numbers which are the two products of 2 consecutive primes and have a 0 in them. 2021 is one of them. What is the other one?

3) sum of factors (ii)^**/*****
2021=43×47, two primes.
The sum of the prime factors of 2021 is 1 + 43 + 47 + 2021 = 2112.
That gives a sum consisting of just 2 different digits. What is the next year that has a sum of its factors that is made up of just 2 different digits?

3) sum of prime factors (iii)^***/*****
The sum of the prime factors of 2021 is 1 + 43 + 47 + 2021 = 2112. 2112 is a palindrome.
What is the next year (product of two primes) in which the sum of its 4 factors is a palindrome?

4) The square of 2012 is 4084441^*****/*****
4084441 consists of 4 different digits. What is the first year in which the square consists of 3 different digits?

5) Bongard problem 2021-1^****/*****

In a Bongard problem, all the pictures on the left share a common property. None of the numbers on the right has this property. What is this property?

6) Bongard problem 2021-2^***/*****

New puzzles are published at least once a month on Fridays. Solutions are published after one or more weeks. You are welcome to remark on the difficulty level of the puzzles, discuss alternate solutions, and so on. Puzzles are rated on a scale of 1 to 5 stars. You can check your solutions here.