Five triangles

Imagine a right triangle (formerly called a rectangled triangle). One side adjacent to the right angle is twice as long as the other. You will not find it very difficult to construct a square from 4 of these triangles.

But is it possible to construct a square from 5 of these triangles? And from 6? From 7?

These problems are derived from ‘doordenkertje’ 12 in the November 1969 issue of ‘Pythagoras’, a Dutch magazine in recreational mathematics. You will need some elementary geometry knowledge to solve this problem.

You can check your solution here

A square

What is the smallest number of which the square ends with three identical digits? And indeed, 0 is excluded as a solution.
This is a slightly simplified version of a problem published as perplexity 466 by Henry Dudeney in The Strand magazine august 1919.

You can check your solution here

Rabbits and carrots

Here is a classic:
If 5 rabbits can eat 5 carrots in 5 minutes, how many carrots are eaten by half as many rabbits in half the time?

This kind of problem goes back to at least the middle ages. They have been formulated in many ways: slaves doing work, painters painting walls, cats catching mice, you name it.

You can check your solution here

Crossnumbers

Crosswords are among the worlds most printed and devised puzzles. And though they are puzzles with words, they are not language puzzles and they can be fairly easily generated once you have a large dictionary in electronic form available.

But Crosswords are so common place I have thus far avoided them in this blog. There are several number variants however, and the one presented here is geared towards math buffs.

Horizontal	Vertical
1 square with identical first and third digit 3 fibonacci number 5 perfect number 7 number of cards in bridge 9 happy number 10 catalan number 11 monodigit number 12 lucas number 14 happy number 16 narcistic number 18 circular prime 19 fibonacci number	1 factorial number 2 prime number 3 fourth power 4 catalan number 6 multiple of 11 8 third power 9 perfect number 12 third power and a square 13 fibonnacci number 15 triangular number 16 fermat prime 17 square

You can check your solution here

The two sisters

Here is another age problem:

On a question about her age, Alice replied:
Me, I’m very old. I’m even older than my sis Barbara and I were together 7 years ago. And though Barbara is just two years younger than I am, she is still very young: she even isnt as old as we were together seven years ago!

What is the age of the girls?

You can check your solution here

My friend and his granddaughter

There are numerous puzzles about ages, and most of them can be solved with elementary algebra, though the hassle of tracking forward and backward into time can sometimes be confusing.

2) My friend and his granddaughter*
A friend told me: 3 years ago, I was thrice as old as my granddaughter. 8 years before I that, that is, 8 years before I was three times as old, I was four times as old.
How old is my friend?

You can check your solution here

Stefan de Banach

The Polish mathematician Stefan de Banach could have said: In the year x I was x^2 years old.

What is the next year in which a person can be born and later make this claim?

You can check your solution here

10 glasses

There are 5 filled and five empty glasses in a row:

A) Can you move 4 glasses in such a way that full and empty glasses alternate?
B) Can you move just 2 glasses so that full and empty glasses alternate?

I invite you to comment on the puzzle, but please do not give the solution. The solution will be given later.

You can check your solution here

How many isosceles trapeziums?

In Mathematics, an isosceles trapezium is a quadrilateral one pair of opposite sides are parallel and the base angles are equal in measure. Alternative definitions are a quadrilateral with an axis of symmetry bisecting one pair of opposite sides, or a trapezoid with diagonals of equal length.

How many isosceles trapeziums do you count in this figure?

Just for the sake of clarity, three triangles in one row would make an isosceles trapezium, as the red one in this example:

You can check your solution at here

Please try to solve the puzzles on your own: your self confidence will grow. You are welcome to remark on the puzzles, and I love it when you comment variations, state wether they are too easy or too difficult, or simply your solution times. Please do not state the soultions – it spoils the fun for others. I usually make the solution available after one or two weeks through a link, which allows readers to check the solution without the temptation to scroll down a few lines before having a go at it themselves.

Alcuins 100-problems

In this blog I mentioned Alcuin of Yorks puzzle collection Propositiones alcuini doctoris caroli magni imperatoris ad acuendos juvenes several times. It was possibly published around 800, and contained 53 problems.

A number of these problems are what we now would term “1 equation with 1 unknown”. Among these are a number which expand on a number, performs a number of calculation operations, and give the answer.
Example:
Nr 2: propositio de viro ambulante in via
A certain man walking in the street saw other men coming towards him, and he said to them: “O that there were so many [more] of you as you are [now]; and then half of half of this [were added]; and then half of this number [were added], and again, a half of [this] half. Then, along with me, you would number 100 [men].” Let him say, he who wishes, How many men were first seen by the man?.” How many men were first seen by the man?

Alcuin gives the solution:
Those who were first seen by the man were 36 in number; double this would be 72. A half of half of this is 18, and a half of this number makes 9. Therefore, say this: 72 and 18 makes 90. Adding 9 to this makes 99. Include the speaker and you shall have 100.

To see how this can be solved with elementary algebra, let’s call the number of men x.
– Then that there were so many [more] of you as you are [now]: x+x=2x
– and then half of half of this [were added]: 2x + 0,5x=2,5x
– and then half of this number [were added]: 2,5x + 0,25x=2,75x
– and adding 1 makes 100.
So 2,75x +1 = 100
2,75x=99
11/4x=99
1/4x=9
x=36

Nr 3: propositio de duobus proficiscentibus
Two men were walking in the street when they noticed some storks. They asked each other, “How many are there?” Discussing the matter, they said: “If [the storks] were doubled, then taken three times, and then half of the third [were taken] and with two more added, there would be 100.” How many [storks] were first seen by the men?

This one leads itself for a second way of solving, working backwards:
– with two more added: 100/2=98
– doubled, then taken three times, and then half of the third taken: like in the previous one, the main problem is in figuring out what Alcuin actually meant.

Nr 4: propositio de homine et equis
A certain man saw some horses grazing in a field and said longingly: “O that you were mine, and that you were double in number, and then a half of half of this [were added]. Surely, I might boast about 100 horses.”
How many horses did the man originally see grazing?

nr 36: propositio de salutatione cujusdam senis ad puerum
A certain old man greeted a boy, saying to him: “May you live, boy, may you live for as long as you have [already] lived, and then another equal
Recreational problems Alcuin 5 Albrecht Heeffer amount of time, and then three times as much. And may God grant you one of my years, and you shall live to be 100.” How many years old was the boy at that time?

Nr 40: propositio de homine et ovibus in monte pascentibus
A certain man saw from a mountain some sheep grazing and said, “O that I could have so many, and then just as many more, and then half of half of this [added], and then another half of this half. Then I, as the 100th [member], might head back to my home together with them.” How many sheep did the man see grazing?

Nr 45: propositio de salutatione pueri ad patrem
A certain boy addressed his father, saying, “Greetings, father!” The father responded, “May you fare well, my son, and may you live three times twice your years. Then, adding one of my own years, you will live to be 100.” How many years was the boy at the time?Nr 46: Propositio Recreational problems Alcuin 6 Albrecht Heeffer A dove sitting in a tree saw some other doves flying and said to them, “O that you were doubled, and then tripled. Then, along with me, you would number 100.” How many doves were initially flying?

nr 48: propositio de homine qui obviavit scholaribus
A certain man met some students and asked them, “How many of you are there in school?” One of [the students] responded to him: “I do not want
to tell you [except as follows]: double the number of us, then triple that number; then, divide that number into four parts. If you add me to one of the fourths, there will be 100.” How many [students] first met the man?

I do not intend to publish the solutions of these problems.

There will be more on this topic in my upcoming e-book on number puzzles.