Skip to main content

Is A smaller or larger than B?

Two pieces of paper, each have an unknown number written on it. You pick one of them and get see the number. Guess if this number is smaller or larger than the other.

Instinct might say that is fifty-fifty to guess right. But actually there is a strategy to do better than 1/2.

In fact, if the numbers are chosen from Uniform(-1, 1), there is a strategy to guess right with probability 3/4. !!

Hint for general strategy: pick a random number and compare.
Hint for Uniform(-1, 1): Focus on 0 which divides the number line into 2 halves. What can you say about the chance that both numbers fall to the left or right of 0? Anything interesting?

Comments

Post a Comment

Popular posts from this blog

641 | 2^32 + 1

Problem: Show that \( 641 | 2^{32} + 1 \). Solution: (From Problem Solving Strategies, Arthur Engel) \( 641 = 625 + 16 = 5^4 + 2^4 \). So \( 641 | 2^{32} + 2^{28} \cdot 5^4 \). Also, \( 641 = 640 + 1 = 2^7 \cdot 5 + 1\). So \( 641 | (2^7 \cdot 5)^4 - 1 = 2^{28}\cdot 5^4 - 1 \). Hence \( 641 | 2^{32} + 2^{28} \cdot 5^4 -(2^{28}\cdot 5^4 - 1) \). QED
Post a Comment
Read more

2n+1 3n+1 squares then 5n+3 not prime

Problem: Prove that if \( 2n+1 \) and \( 3n+1 \) are both squares, then \( 5n+3 \) is not a prime. Solution: Proof by contradiction. Suppose that there is n such that 5n+3 is prime. Note that \( 4(2n+1) - (3n+1) = 5n+3 \). Let \( 2n+1 = p^2 \) and \( 3n+1 = q^2 \), where \( p, q > 0 \). We have \( (2p-q)(2p+q) = 5n+3 \). Since RHS is a prime, we must have \(2p - q = 1 \) and \( 2p + q = 5n+3 \). Solving for \( q \) we get \( 2q = 5n + 2 \). Substituting, we get \( 2q = q^2 + 2n + 1 \), or \( -2n = (q-1)^2 \). Since RHS is \( \geq 0 \), we can only have equality when \( n= 0 \) and \( q = 1 \). In this case, we have \( 5n + 3 = 8 \) not a prime. QED.
Post a Comment
Read more

12 coins

Problem: There are 12 identical coins. There might be one that is a counterfeit, which is either heavier or lighter than the rest. Can you identify the counterfeit (if any) using a balance at most 3 times? Solution: I think this is one of the most difficult variations of the coins weighing problem. The main idea is to "mark" the coins after weighing them. Step 1. Divide the coins into groups of 4. Weigh the first two. If they have the same weight, go to step 4. Otherwise, mark the coins belonging to the heavier group 'H', lighter group 'L', and the unweighted ones 'S'. We know that the counterfeit is either in L or H. Step 2. Now we have 4 Hs, 4 Ls, and 4 Ss. We now form 3 groups: HHL, LLH, and HLS. Step 3. Weigh HHL against HLS. Case 3.1: HHL is lighter than HLS. Then either the L in HHL or the H in HLS is the counterfeit. Simply weigh one of them against S and conclude accordingly. Case 3.2: HHL is heavier than HLS. Then the counterfeit ...
Post a Comment
Read more