Durtles

Every line going through the center of a square bisects its area. Does every line going through the center of an equilateral triangle bisect its area too?

All straight lines that bisect the area of a circle go through its center.

A circle of radius 5 has chord AB with length 6.  A second chord is drawn here and is bisected by AB.

The number 6 has two pairs of factors: 2x3 and 1x6.  These two pairs of factors are called friendly because the sum of one pair is equal to the difference of the other pair, ie. 2+3 = 6-1 = 5.

Let’s remove the central portion of a regular tetrahedron such that what remains are 4 tetrahedrons each with half of the side length of the original.

Consider a natural number N and the sum of its digits S.  Let’s call a divisor of S a Digit Sum Divisor of N.

We have a bracelet made of yellow and green beads, and we want to divide each coloured beads evenly among friends with the least number of cuts.

In this 4x4 grid, if we choose two squares, for which pairs of squares can we find a path from one square to the other, moving only vertically or horizontally at each step, and visiting every square in the grid exactly once?

From the list of numbers 1 to n, cross out every other number starting with the smallest and repeat the process with the remaining numbers until only one number is left.  What number would that be for a given n?

If the possible fairness of a coin is a continuum, the probability of the coin being exactly fair (50H/50T) is 0.  To begin exploring, let’s say the coin can only be rigged heads (100H/0T), fair (50H/50T), or rigged tails (0H/100T).

Here’s an arrangement of the numbers 1 to 4, and what it looks like plotted into a 4x4 grid such that each row corresponds to a number in the arrangement.  Notice that the 3 dots on the bottom right form a straight line.

Here’s a way to put 6 dots into a 3x3 grid such that no 3 dots form a straight line.

There’s a combination lock with a 3 digit code with no repeated digits, and for each of our attempts to guess the code it tells us the number of correct digits and ones in the correct places.

If we draw a line through the center of a square, we can fold the square in half along this line.  Depending on the angle this line makes with the horizontal, as shown in the diagram, the folded square will have different looks.

Here’s one way to put a pair of 1s, a pair of 2s, and a pair of 3s in a row such that the pair of 1s have one number between them, the pair of 2s have two numbers between them, and the pair of 3s have three numbers between them:

For any point inside a square, the sum of the distances from this point to the four sides is half of the perimeter.

Let’s draw a smaller square inside a unit square by dividing out a fixed fraction of each side and connecting the dividing point to an opposite angle.

Starting with 3, let’s double it and add one (or 2n+1), then modulate the answer by 14 (divide by 14 and take the remainder).  If we repeat these two steps, we get the first cycle.

Starting with a natural number at the top, we can split it at each level such that the two lower neighbours always add up to the higher neighbour.  Let’s repeat this process until a row can’t be split using whole numbers in the following row anymore.

In the following puzzle, each number indicates the distance (vertically, horizontally, or a combination of both) this tile needs to move.  A solution is a rearrangement of all the tiles in the grid that satisfies all the numbers where none of the tiles overlap.

Consider the set of natural numbers 1 to 8.  For this set of numbers, 6 is called the balance because the numbers before 6 add up to the same sum as the numbers after 6.  In other words, 1+2+3+4+5 = 7+8.

Consider the subtraction game that goes like this: Write down one whole number at each corner of the square below, then fill in each side with the positive difference the two neighbouring corners and connect the four new numbers in a smaller square.  Repeat the subtraction to continue forming new squares in the middle.

The original problem asks for a combination of 1x2 and 1x3 rods that can tile a 5x5 square. Tiling problems have a tendency to get complicated quickly, so let’s start small.

We start with a polygon and a number of points inside it.  We are asked to draw line segments between the points to subdivide the entire polygon into triangles.

The original problem asks to cut a square by drawing straight lines between any number of midpoints of sides and corners of the square such that a resulting region is 1/5 of the area of the whole square.

Imagine instead of a two dimensional grid, we are now filling a 3D grid made up of nxnxn unit cubes with numbers 1 to n such that every row, column, and depth has exactly one of each number.

This problem is inspired by this geometry puzzle on Twitter. The original problem asks to find the ratio between two equilateral triangles fitted into a square.  Let’s generalize it.

Today let’s explore one of the deviations from the standard sudoku rules that every row, column, and subgrid (excluded so we can generalize to non-perfect-square side lengths) must have exactly one of each number from 1 to n, where n is the side length of the sudoku puzzle.  Let’s include the new rule that a valid solution to a sudoku puzzle must also have one of each number in each of its two diagonals.

Now let’s consider the surfaces we obtain by stretching and gluing the sides of a square together.

Sudoku is a one-player game where the player tries to fill the spaces of an nxn grid using numbers 1 to n, where n is a natural number, such that every column and every row contains exactly one of each number.

Let’s define polycubes as three dimensional shapes made up of unit cubes connected by shared faces.

Let’s consider tiling square grids using polynominoes of any number of units.

If we separate all tetrominoes according to their perimeters, we would have two groups.

Polynominoes are figures made up of square units connected by edges.

Let’s consider tiling square grids with tetrominoes.  Pieces can be rotated, reflected, and repeated as needed.

Consider drawing squares using the lattice points in a grid as vertices.

Consider trying to scramble a string of numbers, say 1, 2, 3, 4, 5, but with the requirement that none of the numbers can end up where they started.  Let’s define this as a complete scramble.

We want to flatten the outer shell of a cube without tearing or stretching the surface.

For all questions in this problem, let’s ignore the effects of gravity, which means higher unit cubes can float in space without lower unit cubes to support them.

Let’s say instead of “bisecting” the triangle, we draw a line from the top vertex to the base with a length that is the average of the remaining two sides...

Let’s start with a right triangle with a height of 1 unit and consider two ways of dividing it into two triangles.

We choose one space in a rectangular grid made of unit squares, and we count the number of squares or rectangles that cover this space.

Let’s consider all real valued functions whose inverse is the function itself.  Since the graphs of these functions are their own mirror images across the line y=x, let’s define these functions as mirror functions.

Continuing from last week’s problem about counting squares, let’s look at the related problem of counting the number of rectangles in a grid, with extensions

Consider the widely known problem of counting the number of squares in a square grid.  Today we will try to extend the problem.

Consider this family of popular puzzles:  You have two empty water jugs with no markings.  One holds exactly 8 liters of water when full, and the other holds 3 liters.  You also have a full tub with an endless supply of water and another large unmarked empty jug of unknown volume.  The goal is to measure out a given volume of water to put into the large jug using the other two jugs.

Starting with a square, choose two points on the perimeter of the square to make a line, and then move both end points incrementally in the same direction (clockwise or counterclockwise) and the same distance along the perimeter of the square to make subsequent lines.

In a game of Peg Solitaire, 32 pegs are arranged in a cross shaped grid with the center peg removed. In each move, the player is allowed to move one peg over another, vertically or horizontally onto an empty space, and remove the peg that was jumped over. The game continues until no legal move is available, and the goal is to have as few pegs remaining as possible when the game ends. A perfect game would end with a single peg back in the center of the board.

Tower of Hanoi is a one-player game using three vertical rods and a number of disks, all of different diameters.  The goal of the game is to move the entire pile of disks onto one of the other two pegs.

The standard number system we use today is called decimal numbers, or base 10, meaning at every digit, we count to 9 before carrying one to the next smallest digit.

A continued fraction is a way to represent any real number as a sequence of integers by writing down the integer part of the real number and taking the reciprocal of the remaining part at each iteration.

A perfect number is a natural number that is equal to the sum of all of its proper divisors (divisors that are less than the number itself).  For example, 6 is the smallest perfect number because its proper divisors, 1, 2, and 3, add up to exactly 6.

Consider all pairs of natural numbers that add to 9, which pair has the largest possible Least Common Multiple?

Triangular numbers and trapezoidal numbers are both natural numbers that can be expressed as the sum of more than one consecutive natural numbers.

Some numbers can be expressed as the sum of two perfect squares...

Sliding numbered blocks in a square grid to put them back in numerical order.

Constructing an unsolvable setup in a colour sorting game.

Lights connected in a string or a loop, each attached to a switch that will change the state of the light itself and its neighbours when pressed.

Connecting letters on the left to their matching letters on the right without crossing any lines.

乌龟的速度到底比兔子慢多少？

龟兔赛跑里的兔子是家兔还是野兔？它们有什么区别？

The card game Seven is one where players take turn extending the numerical sequences of cards in their respective suits.

小迪闲来无聊，要小丹给他讲关于人类世界的事，于是小丹指了指书架底部的百科全书……

Irish Snap is a multiplayer card game using the standard deck of 52 playing cards where each player takes turns laying down one face up card each while counting.  I.e. the first person says “ace” while laying down the first card; then the second person says “two” while laying down the next card.  This process is repeated until the card that has just been laid down matches the number the person just called.  When such a match happens, all players race to slap a hand on top of the pile of cards, and the slowest person has to take the pile into their hand.  The first person to empty their hand wins.

“再等一下，小燕子，再帮帮我吧。”王子又恳求着。

小王子又见到了当初承诺送他回家的蛇，但飞行员却劝他不要走……

Doing in-shuffles on a deck of n cards.

Doing out-shuffles on a deck of n cards.

小王子在地球上遇见的最后一个人是个长得像大人的飞行员。

小王子在途径六颗小星球后到达了地球……

Cutting three dimensional shapes and looking at the possible shapes of the cross section.

小迪是谁起的绰号？

Each regular polygram has a regular polygon at its center.

怎样才能让变丑的王子逃过被重铸的命运？

Cutting 3D shapes parallel to the base such that the top and bottom resulting portions have the same volume.

艾尔病好后一直在寻着生病时用翅膀为他扇风的燕子。

Is this method of constructing a regular polygon an approximation? or theoretically accurate?

Separating egg-laying chickens from non-egg-laying chickens by keeping them in boxes.

Counting the number of ways to make an exact amount using Canadian coins and bills.

“等等，你怎么又写我死掉的故事？”灰灰生气地说。 “啊，真的！一不留神就又选了这样的故事。”小丹不好意思地挠了挠脸。“不过别急，这个王子还是这个故事的主角呢。”

“后果？……控制？……”小丹眨眨眼睛。“我听糊涂了。第一，所有人都知道，梦里的事都是假的，哪来的什么后果？第二，——”

Counting paths in a square grid.

Tiling a long and narrow gridded strip with square tiles.

小辉不喜欢烟花。烟花会让某个瞬间美得离谱，让人无法放手，却又怎么都抓不住。

小希的感恩节作业没办法完成了，因为她倒霉透了，没有可以感恩的事。

同样的铅笔，为什么在她手里就能画出云丝？为什么她的云能给人在风中飘动的错觉？

灰灰接过布条就在身边的桌面上摊开，然后起身打开台灯，像模像样地读了起来。 “尊敬的……迪安斯特罗萨王子殿下……”

那是这个世界唯一的魔法，可以阻挡整个世界的寒冷。小轩隐约记得，里面有温暖的阳光，有带着花香的微风，有鸟叫声，有干净流动的泉水，还有人对他说： “安心住下吧，我会保护你的。”

从前有一座很高很陡峭的山，叫做飞龙山，山顶上有一个古老的飞龙巢穴。山脚下有个同样古老的村子，叫做翠石村，村民们以偷盗飞龙的宝石为生。村里有个习俗，每个刚满十三岁的孩子，不论男女，都要独自爬上山顶，拿到一颗宝石，然后安全回来。

小丹在梦里遇见了来自童话世界的蟋蟀……

农夫后来怎么样了？饿死了吗？

Hello! This is a self introduction from Durtles.

小丹听说灰灰饿着肚子已经好几天了，可是灰灰吃什么呢？

这是真的，是不可避免的。从小到大，她的每个梦都会成真。

一个下雨天，爱读书的小丹在她习惯独处的小角落里遇到了一个自称是从故事书里掉出来的火柴人。

小丹十分惊讶。火柴人为什么会变成兔子？难道他找到他的故事了？

小丹为火柴人讲述龟兔赛跑的故事，但是火柴人的反应却出乎她的意料。

原来这个故事还有其它版本！小丹还是头一次听说！

小丹给故事写了个喜欢的结局，然后神奇的事情发生了……

孔融看着粉丝转述的父亲的回忆，细细品着字里行间透着的自豪。其实这并不像父亲所说的，是个关于分享和谦让的故事，而是一个关于与人相处和说话方式的故事，但这一点只有孔融自己知道。 就当是细节，忽略不计吧。（687字）

A few problems about sharing apples

Moon over Water A rough painting with crayons and pencil crayons on lined A4 paper, of full moon in a clear sky over still water.

A problem about sorting a shuffled set of cards back into the original order

A question about arrangement of coloured chips on a Connect Four board

小美人鱼的头发让她每天都很心烦。她决定去求女巫把她变成一条鱼。（298字）

Four problems about splitting a natural number into smaller natural numbers.

A thought experiment of shuffling a deck of n cards.