#100factorial

Numbers of Letters

~ 01 January 1999 (Source)

The numbers seven, eleven, fifteen, nineteen make a sequence where the numbers go up by 4 each time. But if you count the number of letters in each word, you get a sequence of numbers that goes up by 1 each time – 5, 6, 7, 8.

The list above has 4 numbers in it.

Find a list with 6 numbers in it.

(That is, you need 6 numbers that go up by the same amount each time, but the number of letters is an arithmetic sequence too.)

One Hundred Factorial

~ 01 January 1999 (Source)

The number 100! (pronounced aloud as “one hundred factorial”) is the number produced when all the numbers from 1 to 100 are multiplied together.

That is, 100! = 1 × 2 × 3 × … × 99 × 100.

When this number is calculated and written out in full, how many zeroes are on the end?

Only Ones

~ 01 January 1999 (Source)

Using any or all of the operations of addition, subtraction, multiplication, division and exponentiation (and brackets), and as many of the number 1 as you need, produce each of the numbers from 2 to 20. For example, here is one way to make the number 17:

$(1 + 1 + 1)^{(1+1)} \times (1 + 1) − 1 = 17$

What is the smallest number of 1’s needed to make each number from 2 to 20?

(Note you can’t concatenate the 1’s to make numbers like 11 – each 1 must stand alone as its own number.)

Red and Black

~ 01 January 1999 (Source)

An ordinary pack of 52 cards is arranged so that the black and red cards alternate. You cut the deck and do a single riffle shuffle.

Before you began shuffling you noticed that the cards on the bottom of the two piles were different colours. When you’ve finished the shuffle, you begin laying out pairs of cards from the top of the pile.

What’s the probability that every pair of cards that you lay out has both a red and a black card?

Sum and Product

~ 01 January 1999 (Source)

What 3 numbers have the same product and sum?

Ten From Primes

~ 01 January 1999 (Source)

Make 10 from 2, 3, 5, 7.

What Number Am I?

~ 01 January 1999

$p(x)$ is a polynomial I’m thinking of that has all non-negative coefficients.

You can tell me a number $k$ and I’ll tell you $p(k)$ . You can do this as often as you want.

How many times do you have to ask for $p(k)$ before you can know $p(x)$ ?

Pi on the Floor

~ 16 March 2016 (Source)

Using any of the operations of +, -, *, /, as many of the number π as you need, and as many of the floor function ⌊·⌋ as you need, make each of the whole numbers from 1 to 20. What is the least number of π’s required to make each number?

Unequal Stacks

~ 07 June 2016 (Source)

Start with two unequal stacks of counters.

Move counters off the larger stack to double the size of the smaller stack.

Carry on this way.

What happens?

A stack of 7 and a stack of 2 are one way to start with 9 counters.
Choose other ways.
What happens?

What happens if you start with a different number of counters?

Unequal Stacks image

n to the Five

~ 16 September 2016 (Source)

Is $n^5 + 5^n$ ever prime?

Function Fun

~ 23 September 2016 (Source)

Suppose

$f(3) = \sqrt(3)$ $f(-a) = -f(a)$ $f(a+b) = (f(a)+f(b))/(1-f(a)f(b))$

Evaluate

$f(1)f(2)f(3)f(4)$

Consecutive Integers

~ 30 September 2016 (Source)

Three consecutive integers are multiplied together, and the middle number is added to the total.

E.g. $(4 \times 5 \times 6)+5=125=5^3$

Prove that this is always true, with any set of three consecutive integers.

Making Forty

~ 03 May 2017

Use all of these symbols and only these symbols to produce the number 40:

(())xxx+++3331111

Extension

What is the largest number that can be made using all those symbols?
What is the smallest number that can be made using all those symbols?
What is the smallest odd number that can be made with those symbols?

The End of One Hundred Factorial

~ 09 May 2017

What numbers can be the number of zeros on the end of a factorial?

Bookended Sixes

~ 19 July 2017

Find the smallest number n, such that n ends in a 6, and when n is multiplied by 4 it makes a number that is n with the 6 in the front.

Ten Statements

~ 06 September 2017

Below are ten statements concerning X, a whole number between 1 and 10 (inclusive). Not all of the statements are true, but not all of them are false either. What number is X?

X equals the sum of the statement numbers of the false statements in this list.
X is less than the number of flase statements in this list, or statement 1 is false, but not both.
There are exactly three true statements in this list, or statement 1 is false, but not both.
The previous three statements are all false, or statement 9 is true, or both.
Either X is odd, or statement 7 is true, but not both.
Exactly two of the odd-numbered statements are false.
X is the number of a true statement.
The even-numbered statements are either all true or all false.
X equals three times the statement number of the first true statement in this list, or statement 4 is false, or both.
X is even, or statement 6 is true, or both.

Remainder One

~ 04 October 2017 (Source)

All the numbers that give remainder 1 will go into the number 1 less than the number.

Jiggly Numbers

~ 11 April 2018

A positive integer is said to be jiggly if it has four digits, all non-zero, and no matter how you arrange those digits you always obtain a multiple of 12.

How many jiggly positive integers are there?

Extension

What about varying number of digits?

Zero Zeros

~ 25 July 2018

Use only the five numbers 10, 100, 1000, 10 000, 100 000, with each appearing exactly once, and as many of +, – , ×, ÷ (and brackets) as you like to make numbers with no zeros in their digits.

Broken Calculator

~ 10 August 2018 (Source)

Imagine your calculator is broken. Although it will still display numbers, only the 4 key, + key, = key, and clear key work.

Starting at 0, can you get the calculator to display 1000? What is the fastest way, with the fewest button presses?

Split 25

~ 12 September 2018 (Source)

Take the number 25, and break it up into as many pieces as you want.

$\begin{align*} 25=&10+10+5 \\ 25=&2+23 \\ 25=&1+1+1+1+1+1+1+1+1\\ &+1+1+1+1+1+1+1+1\\ &+1+1+1+1+1+1+1+1 \end{align*}$

What is the biggest product you can make if you multiply those pieces together?

Will your strategy work for any number?

Visual Puzzles

Combo Cube

~ 01 January 1999 (Source)

Eight cubes are marked with one dot on two opposite faces, two dots on two opposite faces and three dots on two opposite faces.

The eight cubes are glued together to form a bigger cube.

The dots on each face of the large cube are counted, to get six totals.

Can the cubes be arranged so that the six totals are consecutive numbers?

One Spot Dice

You have two blank 6-sided dice. You are allowed to choose any number of the twelve faces and draw exactly one dot on each of those faces. Now roll both dice.

Can you draw the dots so that the possible totals are equally likely?

Extension: Put dots on several faces of two 6-sided blank dice.

Can you do it so all totals are equally likely (with more than two possible totals)?

Panda Squares

~ 01 January 1999

Six of Twelve for Twenty One

9 9 9
5 5 5
3 3 3
1 1 1

Quarter the Cross

~ 12 April 2016 (Source)

Colour in one quarter of this cross.

You have to be sure it’s exactly one quarter.

Quarter the Cross image

Herpetomino

~ 23 November 2016

Herpetomino: A flat shape made of squares joined along edges that “looks like a snake”.

Herpetomino image

Seven Sticks

~ 08 August 2017 (Source)

I have seven sticks, all different lengths, all a whole number of centimetres long. I can tell the longest one is less than 30 cm long, because it’s shorter than a piece of paper, but other than that I have no way of measuring them.

Whenever I pick three sticks from the pile, I find that I can’t ever make a triangle with them.

How long is the shortest stick?

Twelve Matchsticks

~ 11 July 2018 (Source)

Twelve matchsticks can be laid on the table to produce a variety of shapes. For example:

                               __
 __ __          __ __ __      |  |
|     |__      |        |     |  |
|__    __|     |        |     |  |
   |__|        |__ __ __|     |  |
                              |__|

Some shapes have the same area and some have a different area.

Arrange 12 matchsticks to make a shape with area 4.

Twelve Matchsticks Image

Dividing Dice

~ 22 August 2018 (Source)

A standard six-sided die spontaneously starts to divide like a living cell. The spots on the die move during the process and spread across the faces of the resulting pair of dice. THat is, the two daughter dice share between them the spots from the original parent die, but possibly in new locations. After a while the new spots are fixed in place.

It turns out that when the two new dice are rolled at the same time, the possible totals are still the numbers from 1 to 6, and they are all still equally likely. How could the spots be arranged on the two dice?

Geometry Puzzles

Squares in a Triangle

~ 09 June 2016 (Source)

For which right triangles is the pink square larger? For which right triangles is the blue square larger?

Squares in a Triangle image

The Central Rhombus

~ 13 June 2016 (Source)

In a rectangle, the midpoint of two opposite sides are marked, and the corners of the rectangle are folded to meet the midpoint of the opposite side. This draws four lines, which mark out a rhombus in the centre.

Q1: Sometimes the rhombus is outside the confines of the rectangle. For which ratios does this happen?

Q2: Under what conditions is the ratio of the diagonals of the rhombus the same as the ratio of the sides of the rectangle?

See source for a Geogebra diagram.

Squares Inside Rectangles

~ 16 June 2016 (Source)

See source for more examples.

Squares Inside Rectangles image

Angles on a Grid

~ 26 June 2016 $x+y+z=?$

Note: The big square is divided into nine small squares of equal size.

Angles on a Grid image

Spiral Coordinates

~ 31 August 2016

Working on a formula to take point coords and tell the spiral step at #100factorial @Veganmathbeagle @samjshah pic.twitter.com/sCGGDatGKE
— David Butler (@DavidKButlerUoA) August 31, 2016

Shaded Fraction

~ 13 November 2016 (Source)

What fraction is shaded?

Shaded Fraction image

Hexagon in a Circle

~ 05 April 2017

Hexagon with verticies on a circle has three consecutive sides 3 and three consecutive sides 5. What is the area of the circle?

Triangles on a Triangle Grid

~ 03 May 2017

Pick any two points on a triangular grid. Make an equilateral triangle with two corners at those points.

Where is the third point?

Triangles on a Triangle Grid image

Square the Rectangle

~ 04 October 2017 (Source)

Take a rectangle with dimensions 1 x 3.

Make 2 cuts. Rearrange the pieces to create a square.

Unequally Divided Parellelogram

~ 04 April 2018 (Source)

Through a given point within a parallelogram draw a straight line to divide the parallelogram into two parts as unequal as possible.

Unequally Divided Parellelogram image

Oblique Cylinder

~ 11 April 2018 (Source)

What is the net of an oblique cylinder, and what is its surface area?

Straight Line Subtended

~ 11 April 2018 (Source)

E is a point at which two given portions of the same straight line subtend equal angles. What is the locus of E?

Straight Line Subtended image

Four Dots, Two Distances

~ 01 August 2018 (Source)

Find all configurations of four (distinct) points in the place that will determine exactly two distinct (non-zero) distances.

Dudeney Puzzles

A selection of some of the more satisfying puzzles drawn from Henry Ernest Dudeney’s “The Canterbury Puzzles” and “Amusements in Mathematics”.

The Glass Balls

Amusements in Mathematics #270

A number of clever marksmen were staying at a country house, and the host, to provide a little amusement, suspended strings of glass balls, as shown in the illustration, to be fired Pg 79at. After they had all put their skill to a sufficient test, somebody asked the following question: “What is the total number of different ways in which these sixteen balls may be broken, if we must always break the lowest ball that remains on any string?” Thus, one way would be to break all the four balls on each string in succession, taking the strings from left to right. Another would be to break all the fourth balls on the four strings first, then break the three remaining on the first string, then take the balls on the three other strings alternately from right to left, and so on. There is such a vast number of different ways (since every little variation of order makes a different way) that one is apt to be at first impressed by the great difficulty of the problem. Yet it is really quite simple when once you have hit on the proper method of attacking it. How many different ways are there?

The Troublesome Eight

Amusements in Mathematics #399

Nearly everybody knows that a “magic square” is an arrangement of numbers in the form of a square so that every row, every column, and each of the two long diagonals adds up alike. For example, you would find little difficulty in merely placing a different number in each of the nine cells in the illustration so that the rows, columns, and diagonals shall all add up 15. And at your first attempt you will probably find that you have an 8 in one of the corners. The puzzle is to construct the magic square, under the same conditions, with the 8 in the position shown.

Game Puzzles

Prime Climb

~ 01 January 1999

SET

~ 01 January 1999

Spot It

~ 01 January 1999

Puzzlebomb 55

~ 01 June 2016 (Source)

Fill the grid with the numbers 1-35, using the clues given, so that each row contains 7 numbers each with a different remainder when you divide by 7, and each column contains 5 numbers each with a different remainder when you divide by 5.

The numbers in each of the dotted regions add up to the indicated total.

Puzzlebomb 55 image

Sprouts

~ 28 September 2016 (Source)

Set up:

Draw any number of dots.

Rules:

Draw a line between two dots, or a loop from one dot, and create a new dot along the new line.
Lines may not cross other lines.
The last player to draw a line wins.

AtoZ 180723 image

Other Puzzles

Convexity of Deltahedra

~ 12 September 2018

Claim: There is a convex deltahedron with every even number of faces from 4 to 20, except one.

Investigating non-convex deltahedra at #100factorial today. pic.twitter.com/xLlq2c6VmW
— David Butler (@DavidKButlerUoA) September 19, 2018

What is the most the least convex deltahedron you can find?

What is the most convex non-convex deltahedron you can find?

Solutions

Number Puzzles

Lousy Labelling Solution

Pull a single ball out of the box labelled ‘Mixed’. This will tell you all you need to know.

If the colour is red, then the box you pulled a ball from must only have red balls (as we know it can’t be the mixed box). The box that is labelled ‘Blue’ cannot contain only blue balls, and so must contain a mixture of both colours. This leaves the box labelled ‘Red’ to be filled only with blue balls.

The same logic applies if the colour drawn from the box labelled ‘Mixed’ was blue, only the colours red and blue are reversed.

The two arrangements are below are the only possiblities, and pulling from the box labelled ‘Mixed’ tells you which arrangement has occured.

Box	Label
Red	Mixed
Blue	Red
Mixed	Blue

Box	Label
Red	Blue
Blue	Mixed
Mixed	Red

Making Forty Solution

There are a class of solutions around constructing 13, multiplying by 3, then adding 1 to get 40.

$((3 \times 1 + 1) \times 3 + 1) \times 3 + 1 = 40$ $(3 \times (3 + 1) + 1) \times + 1 \times 1 = 40$

Extension

Examples of smallest numbers:

$(1 \times 1) + (1 \times 1) \times 3 + 3 + 3 = 10$ $(1 \times 1 \times 3 \times (1 + 1)) + 3 + 3 = 12$ $1 \times 1 \times 3 \times (3) + (1 + 1) + 3 = 14$ $(3 + 1) \times 3 + (3 + 1) \times 1 \times 1 = 16$ $3 \times 3 + 3 \times (1 + 1 + 1) \times 1 = 18$ $3 \times (3 + 3) \times 1 \times 1 + 1 + 1 = 20$ $3 \times (3 + 3 + 1) \times 1 \times 1 + 1 = 22$ $1 \times 3 \times 3 \times 3 + 1 + 1 + 1 = 30$ $1 \times 3 \times 3 \times (3 + 1) + 1 + 1 = 38$

Examples of smallest odd numbers:

$1 \times (3 + 1) \times (3 + 1) \times 3 + 1 = 49$ $((1 +1 + 1 +1) + 3) \times 3 + 3 = 63$ $(3 + 1 + 1) \times (3 + 1 + 1) \times 3 = 75$

Examples of largest numbers:

$3 \times (1 + 1 + 1 + 1) \times 3 \times (3) = 108$

Jiggly Numbers Solution

Cannot be 0.
12 is even, so every arrangement of the 4 digits must result in an even number, so all 4 digits must be even numbers.
- Options: 2, 4, 6, 8
Must be a multiple of 3
Must be a multiple of 4
100 is divisible by 4, so to check divisibilty by 4, only need to check last 2 digits

Answer: 6 (4488, 4848, 4884, 8448, 8484, 8844)

Zero Zeros Solution

$((1000/100)/10)^1000^100000 = 1$ $( 10 / 1000 + 10000/100000) \times 100 = 1.1$ $(100000/10000 + 1000/100)/10 = 2$ $(1000/100000 + 100/1000) \times 10 = 2$ $10 - (100 \times 100000)/(1000 \times 10000) = 9$ $10 - (100/1000 + 10000/100000) = 9.8$ $(1000 + 100)/10 + 10000/100000 = 11$ $1000 - (10 \times 100000)/(100 \times 10000) = 999$ $(10000 - 10) - (100000/(100 \times 1000) = 9989$ $(100000 - 10000) - (1000/(10 \times 100)) = 89999$

https://twitter.com/DavidKButlerUoA/status/737892392947290117

Visual Puzzles

Geometry Puzzles

Dudeney Puzzles

Game Puzzles

Other Puzzles

Unsorted

Amidakuji

2018

Exploding Dots

Competative Four-colouring

Competitive four-colouring at #100factorial. Two teams. Colour a region so that same colours don't touch by edge. Last valid move wins. pic.twitter.com/F1j0piDuI9
— David Butler (@DavidKButlerUoA) September 20, 2017

Circles in a triangle

Card walk

Polydivisible number

???

Sums of squares

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Buckets of fish!

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Domino Grids

The Crossword that counts itself

Akari

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Open Middle problem

James Tanton

Fraction Skyscraper

Mathematical Card Magic

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Interesting thing to investigate for #100factorial soon https://t.co/IrKg7C3ZAn
— David Butler (@DavidKButlerUoA) August 23, 2016