πΊππ΄ββ οΈπ§ "Weekly Puzzle"
#euclidea_puzzle_30
The task is:
π Searching for a buried #treasure, the Pirate walked 1 mile South, turned East and walked another 1 mile, turned 30 Β° North and walked another 1 mile. At what angle does the Pirate need to turn now to return straight to the starting point?
What you need:
β Carefully read the statement of the problem and look at the picture closely
β Leave your answer in the comments below! π
β Write a proof why your answer is correct (optional)
Like and share with your friends β€οΈβ€οΈβ€οΈ One winner will be selected early next week. A reward for the puzzle is one promo code to unlock levels in the Euclidea app. β Prizes for the correct answers are awarded in Instagram only. We are waiting for your solutions: https://www.instagram.com/euclidea_app/
#Euclidea#Pythagorea#geometry #math #mathgames #moremath #mtbos #edtech #iteachmath #mathematics #proof #mathchat #logic #puzzle #problem #solution #drawing #brainteaser #piratesofthecaribbean #pirate #quiz #quadrangle #braingames

πΈ A small photo report from the Mathematical Festival, which took place on February 9 at the Economic Faculty of Lomonosov Moscow State University in Moscow. Our station 'Pythagorea game' was very popular. Three tables were not enough and children enthusiastically solved problems standing all around and even on the floor. Their parents and organizers were also curious and then could not stop solving our #puzzles. Thanks for the great festival!π #math#mathgame#mathfestival#Pythagorea #Euclidea #conference #mathematics #geometry

"Weekly Puzzle"
The task is:
π Construct both tangents to the circle from the point A, using FIVE lines only.
What you need:
β Carefully read the statement of the problem and look at the picture closely
β Solve the problem in the #Pythagorea app (level 25.17) or on squared paper
β Post a screenshot of your solution. Your account should be public so that we can see your post.
β Add hashtags #euclidea and #euclidea_puzzle_29
β Mention @euclidea_app on the screenshot
β Write a proof why your answer is correct (optional)
β You can also send us a message with solution in Direct, but publishing a post is better
Like and share with your friends β€οΈβ€οΈβ€οΈ One winner will be selected early next week. A reward for the puzzle is one promo code to unlock levels in the Euclidea app. β Prizes for the correct answers are awarded in Instagram only. We are waiting for your solutions: https://www.instagram.com/euclidea_app/
#geometry#tangent #math #mathgames #moremath #mtbos #edtech #iteachmath #iteachgeometry #mathematics #proof #mathchat #logic #puzzle #problem #solution #drawing #brainteaser #school #quiz

Many - all so far! - of these compositions start like this, in Euclidea.

"Weekly Puzzle"
The task is:
π Construct the third segment with the ends at grid nodes, passing through the intersection point of two given segments.
What you need:
β Carefully read the statement of the problem and look at the picture closely
β Solve the problem in the #Pythagorea app (like level 1.1) or on squared paper
β Post a screenshot of your solution. Your account should be public so that we can see your post.
β Add hashtags #euclidea and #euclidea_puzzle_28
β Mention @euclidea_app on the screenshot
β Write a proof why your answer is correct (optional)
β You can also send us a message with solution in Direct, but publishing a post is better
Like and share with your friends β€οΈβ€οΈβ€οΈ One winner will be selected early next week. A reward for the puzzle is one promo code to unlock levels in the Euclidea app. β Prizes for the correct answers are awarded in Instagram only. We are waiting for your solutions: https://www.instagram.com/euclidea_app/
#geometry#solidgeometry #math #maths #mathgames #moremath #mtbos #edtech #iteachmath #mathematics #proof #mathchat #msmath #hsmath #mathisbeautiful #logic #puzzle #problem #solution #drawing #brainteaser #school

"Weekly Puzzle"
The task is:
π Construct a right #triangle with the given #hypotenuse AB so that one its leg is twice longer than the other (BC = 2AC). Try to find the shortest solution using #compass and #straightedge.
What you need:
β Carefully read the statement of the problem and look at the picture closely
β Solve the problem in the #Euclidea app (like level 1.2) or in the browser version: https://www.euclidea.xyz/game/PerpBisector
β Post a screenshot of your solution. Your account should be public so that we can see your post.
β Add hashtags #euclidea and #euclidea_puzzle_27
β Mention @euclidea_app on the screenshot
β Write a proof why your answer is correct (optional)
β You can also send us a message with solution in Direct, but publishing a post is better
Like and share with your friends β€οΈβ€οΈβ€οΈ One winner will be selected early next week. A reward for the #puzzle is one promo code to unlock levels in the Euclidea app. β Prizes for the correct answers are awarded in Instagram only. We are waiting for your solutions: https://www.instagram.com/euclidea_app/
#geometry #math #maths #mathgames #iteachmath #mathematics #proof #mathchat #msmath #hsmath #instamath #logic #puzzle #problem #solution #drawing #brainteaser

"Weekly Puzzle" #euclidea_puzzle_26
The task is:
π Point H is the orthocenter of triangle CBA. Find the angle BCA, if AC = 2HB and BC = 3AH.
What you need:
β Leave your answer in the comments below! π
β Write a proof why your answer is correct (optional)
Like and share with your friends β€οΈβ€οΈβ€οΈ One winner will be selected early next week. A reward for the puzzle is one promo code to unlock levels in the Euclidea app. β Prizes for the correct answers are awarded in Instagram only. We are waiting for your solutions: https://www.instagram.com/euclidea_app/
#euclidea#geometry#solidgeometry#math #maths #mathgames #moremath #mtbos #edtech #iteachmath #mathematics #proof #mathchat #msmath #hsmath #mathisbeautiful #instamath #logic #puzzle #problem #solution #drawing #brainteaser #highschool #orthocenter #triangle

#euclidea#euclidea_puzzle_25@euclidea_app
The area of BEDF is 2 by a simple angle/length chase:
1. <ADF = <CDE = <CBA = 45 degrees by exterior angle is equal to interior opposite angle in cyclic quadrilateral ABCD
2. <CED = <AFD (given data)
3. CED ~ AFD by 1, 2 and the equiangular criterion
4. <ECD = <DAF by 3
5. <DAF = <BCD by exterior angle is equal to interior opposite angle in cyclic quadrilateral ABCD
6. <BCD = 180 degrees - <ECD by angles in a straight line
7. <ECD = 180 degrees - <ECD by 4, 5, 6
8. <ECD = 90 degrees by 7
9. <ECD = <BCD = <BAD = <DAF = 90 degrees by 4,5,6,8
10. <CED = <DFA = 45 degrees by 2,9 and angle sum of a triangle is 180 degrees
11. <EBA = <BEA = <ADF = <DFA = 45 degrees by 1,10
12. BAE and ADE isosceles triangles by 11
13. AB=AE, AD=AF by 12
14. BD diameter of ABCD by 9 and Thales theorem
15. BD=2 by diameter = 2 x radius
16. AD=sqrt(4-BA^2) by Pythagoras theorem, 14, 15
17. |ABE| = BA^2/2 by area of right isosceles triangle and 13
18. |ADF|=2-BA^2/2 by area of right isosceles triangle, 13, 16
19. |BEDF|=|ABE|+|ADF|=2 by 17 and 18
Thus, the area is 2.

"Weekly Puzzle"
The task is:
π Given a quadrilateral ABCD inscribed in a circle of radius 1. The angle ABC is 45Β°. E and F are the intersection points of the opposite sides of the quadrilateral. It appeared that β BFC = β BEA. Find the area of the quadrilateral FBED.
What you need:
β Carefully read the statement of the problem and look at the picture closely
β Post a screenshot of your solution. Your account should be public so that we can see your post.
β Add hashtags #euclidea and #euclidea_puzzle_25
β Mention @euclidea_app on the screenshot
β Write a proof why your answer is correct (optional)
β You can also send us a message with solution in Direct, but publishing a post is better
One winner will be selected early next week. A reward for the puzzle is one promo code to unlock levels in the Euclidea app. β Prizes for the correct answers are awarded in Instagram only. We are waiting for your solutions: https://www.instagram.com/euclidea_app/
#geometry#solidgeometry#math #maths #mathgames #moremath #mtbos #edtech #iteachmath #mathematics #proof #mathchat #msmath #hsmath #mathisbeautiful #instamath #logic #puzzle #trysomethingnew #solution #drawing #brainteaser

"Weekly Puzzle"
The task is:
π 2020 is the 22nd natural number for which sum of the digits is equal to the number of digits (the first such number is 1, the second - 11, the third - 20, etc.) Find the 330th natural number with this property. #euclidea_puzzle_24
What you need:
β Leave your answer in the comments below! π
β Write a proof why your answer is correct (optional)
One winner will be selected early next week. A reward for the puzzle is one promo code to unlock levels in the Euclidea app. β Prizes for the correct answers are awarded in Instagram only. We are waiting for your solutions: https://www.instagram.com/euclidea_app/
#geometry#solidgeometry#math#maths #mathgames #moremath #mtbos #edtech #iteachmath #mathematics #proof #mathchat #msmath #hsmath #mathisbeautiful #instamath #logic #puzzle #quiz #solution #drawing #brainteaser #puzzles #euclidea

π Dear friends! π₯³
May all the 12 months of the new year be full of new achievements, many interesting problems to solve and beautiful solutions for you! Thank you so much for being with us.ππ»β€π€ Happy New Year and happy solving! π π π π π ππ ππ ππ ππ ππ ππ ππ ππ ππ ππ ππ ππ ππ π π ΠΠΎΡΠΎΠ³ΠΈΠ΅ Π΄ΡΡΠ·ΡΡ! β
ΠΡΡΡΡ Π²ΡΠ΅ 12 ΠΌΠ΅ΡΡΡΠ΅Π² Π½ΠΎΠ²ΠΎΠ³ΠΎ Π³ΠΎΠ΄Π° Π±ΡΠ΄ΡΡ Π½Π°ΠΏΠΎΠ»Π½Π΅Π½Ρ Π½ΠΎΠ²ΡΠΌΠΈ Π΄ΠΎΡΡΠΈΠΆΠ΅Π½ΠΈΡΠΌΠΈ, ΠΈΠ½ΡΠ΅ΡΠ΅ΡΠ½ΡΠΌΠΈ Π·Π°Π΄Π°ΡΠ°ΠΌΠΈ ΠΈ ΠΊΡΠ°ΡΠΈΠ²ΡΠΌΠΈ ΡΠ΅ΡΠ΅Π½ΠΈΡΠΌΠΈ! Π Π΄Π½ΠΈ - ΡΡΠ°ΡΡΡΠ΅ΠΌ Π΄Π»Ρ Π²Π°Ρ ΠΈ Π²Π°ΡΠ΅ΠΉ ΡΠ΅ΠΌΡΠΈ! Π‘ΠΏΠ°ΡΠΈΠ±ΠΎ, ΡΡΠΎ Π²Ρ Ρ Π½Π°ΠΌΠΈ! ππ»β€π€ Π‘ΡΠ°ΡΡΠ»ΠΈΠ²ΠΎΠ³ΠΎ ΠΠΎΠ²ΠΎΠ³ΠΎ Π³ΠΎΠ΄Π°!
#euclidea#newyear2020#geometry#solidgeometry#math #maths #mathgames #iteachmath #mathematics #mathchat #msmath #hsmath #mathisbeautiful #instamath #logic #puzzle #solution #drawing #brainteaser #happynewyear #marrychristmas #wishcometrue

#euclidea#euclidea_puzzle_23 ok here is a less egregious but still horribly inefficient (20e) solution that at least optimizes the method i tried to use before.

@euclidea_app My solution for #euclidea 's #euclidea_puzzle_23 .
AC = 6AB, so BC = 5AB. (3E)
BD = ABΒ² / 2BC = 0.1AB, using aΒ² / 2b by similar iscoceles triangles and dropping a perpendicular by two circles BA and CB. (2E)
BE = 2 (BDΒ² / 2AB) = 0.01AB, using aΒ² / b by similar iscoceles triangles. (2E)
AF = 2 (AB + BE) = 2 (1.01AB) = 2.020AB, by doubling the line segment. (1E)
Total: 8E

@euclidea_app#euclidea#euclidea_puzzle_23
9E solution using 9 compass moves only.
1. A circle centered at B passing through A is drawn and intersects ray AB again at D.
2. A circle centered at D passing through A is drawn and intersects ray AB again at E.
3. A circle centered at E passing through A is drawn and intersects ray AB again at F.
4. A circle centered at F passing through B is drawn and intersects ray AB again at G.
5. A circle centered at D passing through B is drawn and intersects ray AB again at H. Call this circle X.
6. A circle centered at G passing through H is drawn and intersects ray AB again at I.
7. A circle centered at I passing through D is drawn and intersects ray AB again at J.
8. A circle centered at J passing through D is drawn and intersects X at K (either intersection works).
9. A circle centered at K passing through D is drawn and intersects ray AB again at C. This is the required point as AC/AB = 2.020.
Proof:
This required a construction of point J a distance 52 away from A, from which a point a distance 1/50 can be constructed from D.
This method of constructing length ratios using circles is fairly well-known and used about log2(n)+2 constructions to get 1/n. The extra move here was required because the second point to get 1/50th of the segment was not present.

πWeekly Puzzleπ
Dear friends, this time the puzzle will stretch for two weeks. Good news: there will be more prizes π We wish you a Merry Christmas and a Happy New Year!
The task is:
π The length of AB is 1. Construct a point C on the ray AB such that the length of AC is equal to 2.020. Try to find the shortest solution using compass and straightedge.
What you need:
β Carefully read the statement of the problem and look at the picture closely
β Solve the problem in the #Euclidea app (like level 1.1 or 4.6) or in the browser version: https://www.euclidea.xyz/game/Sqrt2
β Post a screenshot of your solution. Your account should be public so that we can see your post.
β Add hashtags #euclidea and #euclidea_puzzle_23
β Mention @euclidea_app on the screenshot
β Write a proof why your answer is correct (optional)
β You can also send us a message with solution in Direct, but publishing a post is better
The winners will be selected after New Year holidays. β Prizes for the correct answers are awarded in Instagram only. We are waiting for your solutions: https://www.instagram.com/euclidea_app/
#newyear2020#geometry #solidgeometry #math #maths #mathgames #mtbos #edtech #iteachmath #mathematics #proof #mathchat #msmath #hsmath #mathisbeautiful #instamath #logic #puzzle #problem #solution #drawing #brainteaser #happynewyear #2020

The red line would divide in two equal parts the rectangle that misses two corners, each of 2 units of area. But since the upper part has lost 4 more units, to compensate this we need to move this line making the upper part gain 2 units and the lower part losses 2 units. This is acomplished by gaining a triangle of 2 units. #euclidea#euclidea_puzzle_22@euclidea_app

"Weekly Puzzle"
A reward for the puzzle is one promo code to unlock hints in the Euclidea app.
The task is: π Divide the shape into two equal (congruent) pieces.
What you need:
β Carefully read the statement of the problem and look at the picture closely
β Solve the problem on squared paper
β Post a screenshot of your solution. Your account should be public so that we can see your post.
β Add hashtags #euclidea and #euclidea_puzzle_22
β Mention @euclidea_app on the screenshot
β Write a proof why your answer is correct (optional)
β You can also send us a message with solution in Direct, but publishing a post is better
One winner will be selected early next week. β Prizes for the correct answers are awarded in Instagram only. We are waiting for your solutions: https://www.instagram.com/euclidea_app/
#geometry#solidgeometry#math #maths #mathgames #mathfun #mtbos #edtech #iteachmath #mathematics #proof #mathchat #msmath #quiz #logicpuzzles #thinking #logic #puzzle #algebra #solution #drawing #brainteaser #solvethis

"Weekly Puzzle"
A reward for the puzzle is one promo code to unlock levels in the Euclidea app.
The task is:
π Given a circle with center at O and a line which are intersected at points Π and B. Construct a point C such that the angle between OC and AB is 45Β° using just 3 compass and straightedge moves. The point C may lie or lie not on AB.
What you need:
β Carefully read the statement of the problem and look at the picture closely
β Solve the problem in the #Euclidea app (like level 1.7) or in the browser version: https://www.euclidea.xyz/game/SquareInCircle
β Post a screenshot of your solution. Your account should be public so that we can see your post.
β Add hashtags #euclidea and #euclidea_puzzle_21
β Mention @euclidea_app on the screenshot
β Write a proof why your answer is correct (optional)
β You can also send us a message with solution in Direct, but publishing a post is better
One winner will be selected early next week. β Prizes for the correct answers are awarded in Instagram only. We are waiting for your solutions: https://www.instagram.com/euclidea_app/
#geometry#solidgeometry #math #maths #mathgames #moremath #mtbos #edtech #iteachmath #mathematics #proof #mathchat #instamath #logic #puzzle #problem #solution #drawing #brainteaser #quiz #quizinstagram #angle45

2 dodecagons #euclidea#euclidea_puzzle_20@euclidea_app
Proof of maximality:
Note that this problem is equivalent to constructing the shortest polygonal chain with k edges. We can WLOG that these chains link up in a clockwise manner.
Now, the sides of the k segments are directed segments which cannot be parallel but can be antiparallel, and must be as short as possible to fit in the n x n grid. The heuristic for this is to generate the shortest k segments that satisfy this at the origin, and then translating them to form the polygonal chain. By ensuring the segments sum to 0, we ensure that the polygonal chain is closed.
To check this sum is 0, we use the Minkowski metric (aka Taxicab or Manhattan metric), in which the distance between two points is defined as the sum of the absolute differences in coordinates between them. This has the interesting property of determining the perimeter of the smallest bounding box around a convex polygon.
Using this, we note that the smallest bounding box for k=12 has a perimeter of 1+1+1+1+2+2+2+2+3+3+3+3=24, which fits within the 7x7 grid - which also has a perimeter of 24. Thus, for n=7, k is at least 12.
We further note that the smallest bounding box for k=13 has a perimeter of 1+1+1+1+2+2+2+2+3+3+3+3+3=27, which doesnβt fit within the 7x7 grid. Thus, a thirteen sided shape or larger will not satisfy being fit in such a grid, so weβre done.

"Weekly Puzzle"
A reward for the puzzle is three promo codes to unlock hints in the Euclidea app
The task is:
π On grid 6x6, construct a convex polygon with maximum number of sides.
The polygon vertices are in the grid nodes.
What you need:
β Carefully read the statement of the problem and look at the picture closely
β Solve the problem in the #Pythagorea app (like level 1.1) or on squared paper
β Post a screenshot of your solution. Your account should be public so that we can see your post.
β Add hashtags #euclidea and #euclidea_puzzle_20
β Mention @euclidea_app on the screenshot
β Write a proof why your answer is correct (optional)
β You can also send us a message with solution in Direct, but publishing a post is better
Three winners will be selected early next week. β Prizes for the correct answers are awarded in Instagram only. We are waiting for your solutions: https://www.instagram.com/euclidea_app/
#geometry#solidgeometry #math #maths #mathgames #mtbos #edtech #iteachmath #mathematics #proof #mathchat #msmath #hsmath #mathisbeautiful #logic #puzzle #problem #solution #drawing #brainteaser #highschool #polygon #convexpolygons #euclid

#euclidea#euclidea_puzzle_19 the first solution uses 12 elementary moves. I put it first as a hint on how to get to 11, so try to find the 11e solution before looking at images 2 and 3 (which are unique but similar 11e solutions). Obviously I'm not competing on this one bc it my puzzle; btw, thanks @euclidea_app !!

Paul Strait and Anders Kaseorg become the authors of a new record in #Euclidea! Almost at the same time, these two smart guys proposed two different versions of a universal solution in 8 E-moves for level 10.4 "Rotation 90Β°". Congratulations! π₯³π€π β
A little #hint:
First, you should construct a point symmetric to the center of rotation A relative to the rotated line BC, and draw circles (A, B) and (A, C). It takes 4 of 8 moves. Then it remains to understand which lines and points will be needed next (it is convenient to use Explore mode for that).
β
#geometry#worldrecord#newrecord #solidgeometry #math #maths #mathgames #moremath #mtbos #edtech #iteachmath #mathematics #proof #mathchat #msmath #hsmath #mathisbeautiful #instamath #logic #puzzle #problem #solution #drawing #sketch #geometria #rotation

"Weekly Puzzle"
The idea of the task belongs to our user @lpstrait
A reward for the puzzle is three promo codes to unlock Night Mode in the Euclidea app.
The task is:
π Inscribe four circles of the same radius to the given circle so that each of them touches two others and the given circle. Try to use as least number moves as possible (the author's record is 11 moves). What you need:
β Carefully read the statement of the problem and look at the picture closely
β Solve the problem in the #Euclidea app (like level 1.7) or in the browser version: https://www.euclidea.xyz/en/game/SquareInCircle
β Post a screenshot of your solution. Your account should be public so that we can see your post.
β Add hashtags #euclidea and #euclidea_puzzle_19
β Mention @euclidea_app on the screenshot
β Write a proof why your answer is correct (optional)
β You can also send us a message with solution in Direct, but publishing a post is better
Three winners will be selected early next week. β Prizes for the correct answers are awarded in Instagram only. We are waiting for your solutions: https://www.instagram.com/euclidea_app/
#geometry#solidgeometry #math #maths #mathgames #moremath #mtbos #edtech #iteachmath #mathematics #proof #mathchat #msmath #hsmath #mathisbeautiful #instamath #logic #puzzle #problem #solution #drawing #brainteaser #highschool

We are going to participate at the OEB Exhibition on technology supported #learning and #training. Berlin, November 27 - 29, 2019. We are excited to meet you at our stand Wintergarden (Hall C) # D37. https://oeb.global/
#Euclidea#geometricproof#OEB2019 #conference #germany #mathematics #mathgames #berlin #geometry #education #edtech #edutech #learningtechnologies #digitallearning #learningconference #OEB19

"Weekly Puzzle"
A reward for the puzzle is three promo codes to unlock hints in the Euclidea app.
The task is:
π Construct the shortest closed path by grid nodes around the circle from point A.
What you need:
β Carefully read the statement of the problem and look at the picture closely
β Solve the problem in the #Pythagorea app (like level 25.4) or on squared paper. It has a similar layout but you need to shift the point A one cell up.
β Post a screenshot of your solution. Your account should be public so that we can see your post.
β Add hashtags #euclidea and #euclidea_puzzle_18
β Mention @euclidea_app on the screenshot
β Write a proof why your answer is correct (optional)
β You can also send us a message with solution in Direct, but publishing a post is better
Three winners will be selected early next week. β Prizes for the correct answers are awarded in Instagram only. We are waiting for your solutions: https://www.instagram.com/euclidea_app/
#geometry#solidgeometry #math #maths #mathgames #moremath #mtbos #edtech #iteachmath #mathematics #proof #mathchat #msmath #hsmath #mathisbeautiful #instamath #logic #puzzle #problem #solution #drawing #brainteaser #highschool