Goldman Sachs puzzles

Find the 4-digit number ABCD such that when multiplied by 4, it gives DCBA (the digits reversed).

Since ABCD × 4 = DCBA and both are 4-digit numbers:

• A must be small (1 or 2), since multiplying a 4-digit number by 4 without carrying into a 5th digit requires A ≤ 2.
• Since ABCD × 4 = DCBA, the leading digit D of the result must equal 4×A's leading contribution, and the last digit of ABCD × 4 must equal A. This means D × 4 must end in A.
• Testing A = 2: D × 4 must end in 2, so D = 3 or D = 8. Since the product DCBA starts with D and equals roughly 4×(2000+) ≈ 8000-9000+, D must be close to 8-9. Testing D = 8 with digits B, C solved by systematic substitution gives ABCD = 2178.

5 rational pirates find 100 gold coins. They must decide how to split it using the following rules: The most senior pirate proposes a distribution. All pirates (including the proposer) vote on it. If 50% or more vote yes, the split happens. If not, the proposer is thrown overboard, and the next most senior pirate proposes. Each pirate wants to maximize their own gold, but prefers throwing someone overboard if their own share doesn't change. How should the most senior pirate split the coins to survive and maximize their share?

Solution (Backward Induction):

Work backwards from the simplest case:

• 2 pirates left (D, E): D proposes [100, 0]. D's own vote (50%) is enough, so D takes all 100, E gets 0.
• 3 pirates left (C, D, E): C needs 2 votes (including their own). E knows that if C is thrown overboard, we're at the 2-pirate case where E gets 0. So C offers E just 1 coin to secure their vote: C proposes [99, 0, 1]. C votes yes, E votes yes (1 > 0) — passes with 2/3.
• 4 pirates left (B, C, D, E): B needs 2 votes. D would get 0 in the 3-pirate scenario, so B offers D just 1 coin. B proposes [99, 0, 1, 0]. B and D vote yes — passes with 2/4 (exactly 50%).
• 5 pirates left (A, B, C, D, E): A needs 3 votes (including their own). In the 4-pirate scenario, C would get 0 and E would get 0. A offers C and E just 1 coin each to secure their votes: A proposes [98, 0, 1, 0, 1]. A, C, and E vote yes — passes with 3/5.