## Puzzle #168: Who Gets the Easter Eggs?

This is a delightful and a very counterintuitive puzzle, again from the NSA collection. Original puzzle has been written by Tim C., an Applied Research Mathematician at NSA.

Alice has a dozen cartons, arranged in a 3×4 grid, which for convenience we have labeled A through L:

 A B C D E F G H I J K L

She has randomly chosen two of the cartons and hidden an Easter egg inside each of them, leaving the remaining ten cartons empty. She gives the dozen cartons to Bob, who opens them in the order A, B, C, D, E, F, G, H, I, J, K, L until he finds one of the Easter eggs, whereupon he stops. The number of cartons that he opens is his score. Alice then reseals the cartons, keeping the eggs where they are, and presents the cartons to Chris, who opens the cartons in the order A, E, I, B, F, J, C, G, K, D, H, L, again stopping as soon as one of the Easter eggs is found, and scoring the number of opened cartons. Whoever scores lower wins the game; if they score the same then it’s a tie.

For example, suppose Alice hides the Easter eggs in cartons H and K. Then Bob will stop after reaching the egg in carton H and will score 8, while Chris will stop after reaching the egg in carton K and will score 9. So Bob wins in this case.

Who is more likely to win this game, Bob or Chris? Or are they equally likely to win?

As always, please send your answers as comments within the blog (preferred), or send an e-mail to alokgoyal_2001@yahoo.com. Please do share the puzzle with others if you like, and please also send puzzles that you have come across that you think I can share in this blog.

Happy egg hunting!

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### 4 Responses to Puzzle #168: Who Gets the Easter Eggs?

1. If A has an Easter egg – tie
Else if B has it – Bob wins
Else if C has it -Bob wins
Else if D has it – Bob wins
Else if E has it – Chris wins
Else if F has it -Chris wins
Else if G has it – Bob wins
Else if H has it – Bob wins
Else if I has it – Chris wins
Else if J has it – Chris wins
Else if K has it – Chris wins
Else if L has it – Its a tie

Tie = 11/12C2 + 0/12C2
Bob wins = (4+5+8+9+10)/12C2
Chris wins = (1+2+3+6+7)/12C2

So prob of tie =11/66=1/6
Prob of Bob winning = 6/11
Prob of Chris winning = 19/66

• Suman says:

I don’t think this is entirely correct. If B and E have an egg even that is a tie.

• Got it. My mistake. Thanks for pointing it out.

2. Prakhar says:

I took two cases. H,l and k,l at the very end and got results as b = 11, c=9 for k,l and b=8, c=11 for h,l. With these cases, I predicted bob will win