Category Archives: Mathematics

Matches and rectangles

Combinatorics, Matchsticks, Recreational mathematics

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.

2021

Arithmetic

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.

Quento

Arithmetic

While cleaning up old newspapers in someone’s house I came across a puzzle called “Quento” in the Dutch newspaper “Algemeen dagblad”.

The basics of this puzzle are simple calculations. What sets it a bit apart from other puzzles is that the answers are given, and that you have to find the exercise.

Example:

You have to find a calculation for each of the numbers on the right by going from a number to a + or – sign and on to another number. If necessary, you may add additional +/- signs and numbers, as long as you don’t use the same number or sign twice.

What you may do:

Not allowed is:

Of course there is a website, and an app for both Android and Ipad. Personally I think the exercises are too easy, though no doubt they can be increased by adding size and moving to higher numbers, as shown in our 4th problem. In the app I did see higher numbers, in the sense of multiples of 3, 4, 5 , and so on, but I didn’t see larger sizes, as in our fourth problem. That may be because I didn’t purchase the app and just used the free version.

Quento 2*/*****

Quento 3*/*****

Quento 4**/*****

You can check your solutions at href=”https://justpuzzles.wordpress.com/Solutions to puzzles 501-750/#516″>here.

Four of a kind

Combinatorics

This weeks puzzle I came across in an old issue of ‘Machazine’, dated July 2017.

How many cards should I draw from a double deck of cards to make sure I hold at least one four of a kind?

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.

Four fruits

Algebra

Recently I noticed a lot of elementary algebra represented as pictures. Personally I doubt that packing exercises in this format helps the learning process, but if people like it as a puzzle, here is an example (with a little twist, of course)

1) Four fruits*/*****>/sup>

What do an apple and a cherry cost me?

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.

Six sixes

Arithmetic

Digits and numbers are wonderful toys. The number of problems with them surpasses the imagination. Still, great minds think alike.
In 2013 I posed a classical problem to make as many numbers as possible by combining four fours.

In March I posed the challenge to make all the numbers of 0 to 20 by combining 5 5’s.

1) 6 6’s***/*****
Today’s problem is a sequel: make all the numbers 0 till 20 using 6 6’s. You can use brackets, +,-,*,/, ^and sqrt. You msy use ! for just one of the numbers.
You can check your solutions here and here

When googling, I found the puzzle masters at occupymath have posed the same challenge.

2) 7 7’s***/*****
Of course the problem can be generalized to 7 7’s. Make all numbers 0-24 using seven sevens, brackets, +-*/, no faculty.
You can check your solutions here and here

3) general ***/*****
The next problem is not, as you may have thought, to make all numbers 0-20 with 8 8’s. Rather the question is, can the numbers 0-20 with made with all digits? In base 11, in base 12?
For a start:
0 = (n-n)*(n+…+n)
1 = n/n + (n-n)*(n+…+n)
2 = n/n + n/n + (n-n)*(n+ … + n)
and so on.

Five five’s

Arithmetic


Some of you may know the problem to make all the numbers from 1 to 20 with four fours.
I wrote about it here. It may come as a surprise to you that it is possible to make all the numbers from 0 to 100.

Today’s problem is to make all numbers from 0 to 20 using exactly five fives. For example:
(5*5 – 5*5) / 5 = 0

You can find my solutions here.