Logical and Mathematical Problems

This page presents some of the problems I found very nice.  I like problems with a (very) short, but difficult to find solution.  I solved all these problems, except those where specified.  I am not the author of any of the problems shown here.  The authors written are the persons who told me about that problem, not necessary the author of the problem.
  1. Logical problems
  2. Mathematical problems

Logical Problems

Here are my two favourite logical problems.

John or George?

100 soldiers form a 10*10 square. From every column the tallest soldier is chosen, and from these 10 soldiers, the shortest is chosen. Let John be his name. In the same position than the beginning, from every line the shortest soldier is chosen, and from these 10 soldiers, the tallest is chosen. Let George be his name. Who do you think is taller: John or George?

Romanian version:

Ion sau Gheorghe?

100 de soldati sunt asezati într-un careu 10*10. De pe fiecare coloanã se considerã cel mai înalt soldat, iar dintre acestia 10 cel mai scund. Fie numele acestuia Ion. De pe fiecare linie se considerã cel mai scund soldat, iar dintre acestia 10 cel mai înalt. Fie numele acestuia Gheorghe. Care credeti cã este mai înalt: Ion sau Gheorghe?

The Equator

Suppose surrounding the Equator (supposed circular) with a thread, so that the thread length is equal to the Equator length. Cut the thread in a point and add to the thread 1 metre. Raise up the thread constantly. What do you think, can a cat pass under the thread ?

Romanian version:

Ecuatorul

Se infãsoarã Pãmântul la Ecuator cu o atã (presupunem Ecuatorul circular), astfel cã lungimea atei va fi exact lungimea Ecuatorului. Se taie ata într-un loc si i se adaugã 1 metru. Se ridicã ata în mod constant (la aceeasi înãltime) fatã de Ecuator. Ce credeti, pe sub atã va putea trece o pisicã?

The Three Children

- How old are your children?
- The product of their ages is 36.
- But their sum?
- It is equal to the number of our neighbour's house.
- I need one more information!
-The eldest child plays at piano.

Romanian version:

Cei 3 copii

- Ce vârstã au cei trei copii ai tãi?
- Produsul acestor vârste este egal cu 36.
- Dar suma?
- Este egalã cu numãrul de la casa vecinului.
- Mai am nevoie de o informatie!
- Copilul cel mai mare cânta la pian.

Sursa: Solomon Marcus - "Controverse in stiintã si inginerie"

Square fill 1

What do you think, can the table below (10*10 squares, but without the two corners) be filled completely with non-superposed 2*1 pieces (shown at the right at the image)?

Square fill 2

What do you think, can the table below (2n*2n squares, but without the one corner square) be filled completely with non-superposed pieces shown at the right at the image?

Division

Divide the following figure in four congruent (equal, identical) parts.

The mirror

Why does the mirror reverse the left-right direction and not another direction?
I didn't solve this problem.

Romanian version:

Oglinda

De ce oglinda inverseazã stânga-dreapta si nu altã directie ?

Sursa: Dragos Apostol

The torus

Suppose an empty torus. Make a little hole on its surface. What do you think, can the torus be reversed by this hole?
I didn't solve this problem.

Romanian version:

Torul

Presupunem un tor (covrig) gol pe dinãuntru. Se înteapã torul într-un punct oarecare. Ce pãrere aveti, se va putea întoarce torul pe dos prin aceastã gaurã?

Sursa: Dan LÃZÃRESCU - "PALEOARITMETICÃ si alte probleme de logicã"

The bridge

Three people (A, B, and C) need to cross a bridge. A can cross the bridge in 10 minutes, B can cross in 5 minutes, and C can cross in 2 minutes. There is also a bicycle available and any person can cross the bridge in 1 minute with the bicycle. What is the shortest time that all men can get across the bridge? Each man travels at their own constant rate.
I didn't solve this problem.

Sursa: Constantin Colescas

Mathematical Problems

Click here to see a nice graphical problem.
Doua poligoane se numesc la fel structurate daca pot fi impartite in poligoane respectiv congruente. Atunci oricare doua poligoane care au aceeasi arie sunt la fel structurate.
I didn't solve this problem.

Choose a positive integer. If it is even, halve it. If it is odd, triple it and add 1. Prove that repeating this procedure you will eventually arrive at 1. It seems that this problem has not yet been solved.
I didn't solve this problem.  Nobody solved until now.

Find all the solutions in N for the following equation:
2x - 13 = 3y

Sursa: Gabriel Simion

Prove that any odd number n which is not divisible by 5, divides a number of the form 111...111.

Sursa: Michel Barret

Find the minim surface which may contain any curve of a given length.
I didn't solve this problem.

Sursa: Olivier Buffet

What's the intersection between a cone and an oblique plan?

The Path Problem

Show that a path on an m by n square grid which starts at the north-west corner, goes through each point exactly once and ends at the south-east corner divides the grid into two equal halves:
(a) those regions opening north or east;
and
(b) those regions opening south or west.
I didn't solve this problem.

Proposed by David Ficher, Karen Collins & Lucia Krompart in "The American Mathematical Monthly", Volume 101, Number 8/October 1994

Written by Eugen Dedu