Mathalon

Math, but multiplayer. Daily problems, live Rush races, ranked 1v1 — rated like chess. Free to play →

[Mathalon @Mathalonapp]

Drop any point inside any triangle. Its distances to the three corners always add up to at least twice its distances to the three sides. Equality only at the centre of an equilateral triangle. The Erdős–Mordell inequality (1935/1937). #mathalon #math #geometry

[Mathalon @Mathalonapp]

Draw any triangle. From each corner, walk along the edges until you're exactly halfway around, and draw a line to that spot. The three lines always meet at one point: the Nagel point (von Nagel, 1836). #mathalon #math #geometry

[Mathalon @Mathalonapp]

Pick any number of corners, spaced evenly around a circle. From one corner, multiply the lengths of all the lines to the others: you always get the number of corners. Roger Cotes, 1722. #mathalon #math #geometry

[Mathalon @Mathalonapp]

Draw any triangle and pick any point on a side. Walk parallel to another side, then always parallel to the side you just left. Six steps and you're back at the start. Thomsen's theorem. #mathalon #math #geometry

[Mathalon @Mathalonapp]

Drop two identical rulers any way you like. Join matching numbers and mark the middle of each band: the midpoints always land on one straight line, evenly spaced. Hjelmslev's theorem. #mathalon #math #geometry

[Mathalon @Mathalonapp]

Draw any three overlapping circles. Join the two crossing points of each pair: three lines, and they always meet at one point, however you move the circles. Gaspard Monge proved it. #mathalon #math #geometry

[Mathalon @Mathalonapp]

Shine sunlight into any round cup. The bright curve at the bottom is a nephroid: a wheel rolling around a circle half the cup's size draws the exact same line. Huygens, 1678. #mathalon #math #geometry

[Mathalon @Mathalonapp]

Put a queen on any square. Players slide her left, down or diagonally; whoever reaches the corner wins.

The losing squares hug two straight lines, and their slope is the golden ratio, 1.618. Wythoff's game, 1907.

#mathalon #math #goldenratio

[Mathalon @Mathalonapp]

Drive around any loop that never crosses itself and add up every turn. However wild the loop, you end at exactly +360°. Make it cross once, like a figure eight, and you get 0°. Heinz Hopf, 1935. Math, but multiplayer: https://t.co/l6s17PLVfL

#mathalon #math #geometry #topology

[Mathalon @Mathalonapp]

Pick any positive fraction. It's already in this tree, exactly once.

Rule: a/b has children a/(a+b) and (a+b)/b. Start at 1/1. (Calkin & Wilf, 2000)

Our first live tournament is today, 16:00 UTC: https://t.co/n0ORfDKSsJ
#math #mathalon

[Mathalon @Mathalonapp]

Paint every point of the plane red or blue, any way you like.

Somewhere two points exactly 1 apart get the same colour. Proof: a triangle with sides 1 has 3 corners and only 2 colours.

Tournament tomorrow: https://t.co/n0ORfDKSsJ
#math #mathalon

[Mathalon @Mathalonapp]

Draw any convex shapes, as many as you like.

If every three of them share a point, then all of them share a point. Pairs are not enough.

Helly's theorem, 1913.

Tournament Sun Nov 1: https://t.co/n0ORfDLqih
#math #mathalon

[Mathalon @Mathalonapp]

Take any 52-card deck. Cut it exactly in half and weave the halves perfectly, top card staying on top.

After 8 perfect shuffles every card is back where it started. A 50-card deck needs 21. The faro shuffle.

Tournament Sun Nov 1: https://t.co/n0ORfDLqih
#math #mathalon

[Mathalon @Mathalonapp]

Pick any one-hump rule, feed each answer back in, turn up the dial: 1 answer, then 2, 4, 8, then chaos.

The gaps between splits shrink by about 4.669 each time. A sine wave gives the same number. Feigenbaum, 1975.

Tournament Sun Nov 1: https://t.co/n0ORfDKSsJ
#math #mathalon

[Mathalon @Mathalonapp]

Draw any loop that never crosses itself. Is a dot inside or outside?

Draw a straight line out and count the crossings. Odd: inside. Even: outside. Any direction works.

The Jordan curve theorem, 1887.

Live tournament Sun Nov 1: https://t.co/n0ORfDKSsJ
#math #mathalon

[Mathalon @Mathalonapp]

Count every square yourself. Same four pieces, and a square goes missing.

The long edge isn't straight: slopes 3/8 and 2/5. The sliver between the two shapes is exactly one square. Paul Curry, 1953.

Live tournament Sun Nov 1: https://t.co/n0ORfDLqih
#math #mathalon

[Mathalon @Mathalonapp]

Color every line between six dots red or blue, any way you like. A one-color triangle always appears.

Five dots can dodge it. Six can't. All 32,768 colorings checked. Ramsey's theorem, 1930.

Live tournament Sun Nov 1: https://t.co/n0ORfDKSsJ
#math #mathalon

[Mathalon @Mathalonapp]

Pick any point in a square. One line, drawn without lifting the pen, passes right through it.

Draw a U, shrink it, copy it 4 times, join the ends, repeat forever. Peano 1890, Hilbert 1891.

Live tournament Sun Nov 1: https://t.co/n0ORfDKSsJ
#math #mathalon

[Mathalon @Mathalonapp]

Stick any five pins into an orange. You can always cut it in half with four of them in one half (pins on the cut count for both). Pick two, cut through both: the other three split, so one side gets two. 2 + 2 = 4.

Live tournament Nov 1: https://t.co/n0ORfDKSsJ
#math #mathalon

[Mathalon @Mathalonapp]

Pick any prime and divide it by 4. Remainder 1? It's always a sum of two squares: 13 = 4 + 9, 997 = 6² + 31². Remainder 3? Never (Fermat, 25 Dec 1640).

Live tournament Sun Nov 1, 16:00 UTC: https://t.co/n0ORfDKSsJ
#math #mathalon