Bar-Ilan's weekly mathriddle competition, 2024

• Question 1 - The 1224th Digit

  We write the natural numbers in order, one after the other from left to right:

  1234567891011...

  Note, for example, that the digit in the 10th place is 1 and the digit in the 11th place is 0.

  Continuing with this writing as much as needed...

  Which digit will be in the 1224th place in the sequence?

  Topics:

  Number Theory Arithmetic Proof and Example -> Constructing an Example / Counterexample Algebra -> Sequences

• Question 2 - Maximum Length Game

  The game begins with a row of 42 coins, all showing 'Heads' up.

  Each player, in their turn, chooses one of the coins showing 'Heads' up - flips the chosen coin, and also the coin immediately to its right (if one exists).

  If you manage to make all the coins show 'Tails' up on your turn, you win the game!

  At most, after how many turns will the game end?

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  Topics:

  Algebra -> Sequences -> Arithmetic Progression / Arithmetic Sequence

• Question 3 - The Number with the Most Divisors

  Among the positive integers less than 1000, which number has the most divisors?

  Topics:

  Arithmetic Number Theory -> Prime Numbers -> Prime Factorization Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM) Number Theory -> Division

• Question 4 - The Winning Exercise

  Two fair dice (1-6) are rolled, and two calculations are made:

  1. The sum of all 10 visible faces (5 on each die)

  2. The sum of the two top faces of the dice, multiplied by 5

  What is the probability that the result of the first calculation is greater than that of the second calculation?

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  Topics:

  Combinatorics Probability Theory

• Question 5 - Knight's Moves

  A knight moves on an infinite grid. It starts at the point (0,0) and must reach
  the point (5,27). Assuming it moves in the smallest number of steps required,
  how many different possibilities does it have to reach this point?

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  Topics:

  Combinatorics -> Combinatorial Geometry -> Grid Paper Geometry / Lattice Geometry

• Question 6 - The Hidden Number

  Bar and Ilan each chose an integer between 1 and 30.

  Ilan: Is your number double mine?

  Bar: I don't know. Is your number double mine?

  Ilan: I don't know. Is your number half of mine?

  Bar: I don't know. Is your number half of mine?

  Ilan: I don't know.

  Bar: I know what your number is.

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  Topics:

  Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures

• Question 7 - How many triangles?

  How many triangles are there in the picture?

  

  Topics:

  Geometry -> Plane Geometry -> Triangles Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Geometry -> Plane Geometry -> Angle Calculation Number Theory -> Division

• Question 7 - How a Wheel Turns

  All 6 wheels in the diagram rotate as they touch each other without slipping. The diameter of the leftmost wheel is 15.7 cm and it makes 12 revolutions per minute.

  It is known that the smallest wheel makes one revolution per second.

  What is the diameter of the smallest wheel?

  

  Topics:

  Arithmetic Geometry -> Plane Geometry -> Circles Algebra -> Word Problems

• Question 8 - The Disguised Digits

  In the following division exercise, (almost) all the digits are disguised!

  What is the dividend?

  \(\begin{align*} &\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }**\text{ }8**\\ &\overline{* * * * * * *\text{ } *}| * *\text{ } *\\ &\underline{\text{ }\text{ }* *\text{ } *}\\ &\text{ }\text{ }\text{ }\text{ }\text{ }* * * *\\ &\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\underline{\text{ }\text{ }* *\text{ } *}\\ &\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }* * * *\\ &\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\underline{* * *\text{ } *}\\ \end{align*}\)

  Topics:

  Arithmetic Number Theory -> Division Puzzles and Rebuses -> Reconstruct the Exercise / Cryptarithmetic

• Question 9 - Two Argue, the Third Takes

  The entire class is in dispute!

  42 think yes, 43 think maybe, and 36 think no.

  When two people who think differently from each other meet, they both change their position to the third.

  What is the minimum number of meetings that must take place until everyone agrees on the same position?

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  Topics:

  Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures