Earlier today I set up three problems for you from a math competition for Mars school children. By Mars I mean Hungarian.
In the mid-twentieth century, a generation of outstanding mathematicians and physicists from Hungary were humorously called Martians, as their intelligence came from another planet.
Here are the puzzles again with solutions.
1. Curb your enthusiasm (ages 13/14)
On an island, every resident is either half-hearted or enthusiastic. A visitor from a distant country was invited to dinner by a group of 10 residents. After dinner, the visitor asked all 10 members of the group about the number of enthusiastic residents within their group.
She got the following answers: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.
Knowing that the answers of the half-hearted individuals cannot be more than the actual answer, and the answers of the enthusiastic individuals cannot be less than the actual number, determine the number of enthusiastic residents within the group.
Solution 6
Arrange the residents in a row according to their answers: 3 4 5 6 7 8 9 10 11 12. The first thing to note is that no half-hearted person can be to the right of an enthusiastic person, so the half-hearted are on. the left side.
Let’s say there are exactly 3 enthusiastic people. If it was, then the person who said ‘4’ must be enthusiastic, as should everyone to their right, 9 enthusiastic people in total. Contradiction! So we can eliminate the solution is exactly 3 enthusiastic residents.
Let’s say there are exactly 4 enthusiastic residents. Again, this leads to a contradiction, because that would mean there are at least 8 enthusiastic residents. (The people to the right of the person who said ‘4’)
Likewise, it is impossible for there to be 5 enthusiastic residents, as this implies at least 7 enthusiastic residents.
If there are 6 enthusiastic people, there are no contradictions. But if there are 7 or more enthusiastic people, then it is impossible, as it would mean that some of these enthusiastic people give answers of less than 7, so it is impossible.
So there are 6 enthusiastic people.
2. Edgy Logo (ages 11/12)
The task here is to design a 2D logo using only equilateral triangles and squares, each with a side length of 1 cm. The triangles and squares must be glued along their entire sides without any overlap.
Make a logo with a circumference of 13 cm from the following shapes or prove it impossible:
a) A single triangle and a few squares
b) The same number of squares and triangles.
c) Only triangles.
d) Squares only.
Solution
a)
b)
c)
d) It is impossible. When you make an object with only squares, all perimeter values are equal, since each edge must have a corresponding edge on the other side of the shape, as in a and b here.
3. Ax Head Tiles (ages 15/16)
The edges of these identical tiles are quarter circles, and their centers are the points marked. Determine the area of a tile, measured in cm2given that the height of a standing tile is 12cm.
Solution 72
The top of a standing tile is a quarter circle, the center of which is the center of the tile, therefore the radius of the quarter circle is 6 cm, and the corners of the tile are also 6 cm from the center of the tile.
We can dissect the tile by removing the circle segments at the top and bottom and placing them in the empty spaces on the left and right sides of the tile. We get a square with a diagonal of 12 cm. If the side length of the square is s, we have according to Pythagoras’ Theorem: s2 + s2 = 122. So s2 = 72 = the area of the square = the area of the tile.
Today’s riddles are taken from the Dürer competitiona mathematics competition for 10 to 18-year-olds that has been held in Hungary since 2007. If you liked it, there’s a lot more in there Mathematical explorations for young minds.
I’m the author of Think Twice: Solve the Puzzles That (Almost) Everyone Gets Wrong, a collection of counterintuitive puzzles that make you think about thinking—while enjoying the pleasure of being tricked. The questions are not ‘trick’ questions; instead, they reveal our biases and flawed reasoning.
Think twice: solve the simple puzzles (almost) all get wrong. To support the Guardian and Observer, order your copy from guardianbookshop.com. Delivery charges may apply. (In the US the book is called Puzzle Me Twice.)
I’ve been doing a puzzle here on alternate Mondays since 2015. I’m always on the lookout for great puzzles. If you want to suggest one, email me.
