Category Archives: Numbers

Old year – New Year

Numbers

It is always hard to come up with original puzzles, let alone with a puzzle that has to do with Christmas or new year. Any way: best wishes for 2015 to all of you!

Let’s have a simple problem from the orient. Did you notice that medieval sultans always seem to have an ample supply of beautiful daughters? And that they invariably have strange ways to choose their son-in-law?

This one is no exception. His kingdom had an extensive seashore, and he said to four young men interested in the hand of his daughter: Along the coastline I own many fishing boats. But there is something very peculiar with the number of fishing boats:
when the number is divided by 2, the remainder is 1,
when the number is divided by 3, the remainder is 2,
when the number is divided by 4, the remainder is 3,
when the number is divided by 6, the remainder is 5,
when the number is divided by 7, the remainder is 6
when the number is divided by 8, the remainder is 7
when the number is divided by 12, the remainder is 11
when the number is divided by 14, the remainder is 13
when the number is divided by 18, the remainder is 17
when the number is divided by 21, the remainder is 20
when the number is divided by 24, the remainder is 23
when the number is divided by 28, the remainder is 27
when the number is divided by 32, the remainder is 31
How many fishing boats are there in my kingdom?

You can check your solution here

Crossnumbers

Numbers,

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.

Crossnumbers math buffs exercise

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

How many isosceles trapeziums?

Counting, Geometry

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?
Matchsticks 4x4 triangles

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

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

Numbers

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.

How many rhombuses?

Counting, Geometry

In Mathematics, a Rhombus is a figure consisting of 4 lines, all of the same size, and with opposing sides parallel.
Thus:
Rhomboid

How many rhombuses do you count in this figure?
Matchsticks 4x4 triangles

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.