A Puzzle on Leap Frog Jumping

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A Puzzle on Leap Frog Jumping
(With a surprising result) By N.D.Prabhakar

Board: Men placed at grid points

• A jump move involves two adjacent pieces (men) along same row or same column • No cross-wise jump • After a jump, the man must land on a vacant position, and the man who is jumped over gets removed from the board • A piece can jump over only one piece in one move

Red , Yellow and Violet pieces can not jump Green can jump right over blue Blue can jump left over green Blue can jump up over yellow Blue can jump down over violet

Levels
ETC. above Level 3

Level 2
Level 1 Level 0 Level -1 ETC. below

Problems
• Designate one of the rows as Level 0, and rows above that as Levels 1,2,3, etc. • You are allowed to place as many men as you need in any configuration of your choice in Level 0 and below (board potentially infinite) • How many men would you need in order to reach a man to level k using leap frog jumping moves, for k=1,2,3,… ?

Reaching Level 1
• Needs two men

Level 0

Reaching Level 1
• Blue piece jumps over green piece: green shown crossed out

Level 1 Level 0

Reaching Level 2
• Needs 4 men

Level 2 Level 1

Level 0

Reaching Level 2
• 1st move: Blue jumps up over green

Level 2 Level 1

Level 0

Reaching Level 2
• 2nd move: Red jumps right over green

Level 2 Level 1

Level 0

Reaching Level 2
• 3rd move: Red jumps up over blue

Level 2 Level 1

Level 0

Reaching Level 3
• Needs 8 men
Level 3 Level 2 Level 1 Level 0

Reaching Level 3
• In 3 moves we saw before, get the blue piece to level 2
Level 3 Level 2 Level 1 Level 0

Reaching Level 3
• In moves 4 and 5, get the violet and yellow pieces jump left Level 3 Level 2 Level 1 Level 0

Reaching Level 3
• In move 6, yellow pieces jumps up over violet

Level 3 Level 2 Level 1 Level 0

Reaching Level 3
• In move 7, yellow piece jumps up over blue

Level 3 Level 2 Level 1 Level 0

Reaching Level 4
• Need 20 men
Level 1 Level 0 Level -1 Level -2 Level -3

Reaching Level 4
• In moves 1 to 5, jump up the blue men in level -1 over the green ones in level 0 Level 1 Level 0 Level -1 Level -2 Level -3

Reaching Level 4
• In move 6, jump the blue man in level 0 left over the green one in level 0 Level 1 Level 0 Level -1 Level -2 Level -3

Reaching Level 4
• In moves 7 and 8, jump up the blue man in level -3 over the green ones in level-2 Level 1 Level 0 Level -1 Level -2 Level -3

Reaching Level 4
• In moves 9 and 10, jump the red men left and right in level -2 over the yellow ones in level-2 Level 1 Level 0 Level -1 Level -2 Level -3

Reaching Level 4
• In moves 11 and 12, jump up the red men in level -2 over the blue ones in level-1 Level 1 Level 0 Level -1 Level -2 Level -3

Reaching Level 4
• We now have same config in level 1 and below as we had in level 0 and below for reaching level 3. So in 8 more moves, we can get one man to reach level 4

Level 1

Level 0

Reaching Level 5
• Impossible with any finite number of men!! • Has a very clever and elegant proof • Fix a position P on level 5 and we show there is no config in levels 0 and below from which one can reach P in finite number of jump moves • We assign a value to each position on the board, with P having value 1 and all others having a value of the type xn where n is the smallest number of steps to reach P • x is chosen to be a number between 0 and 1

Value Assignment
Level 6 Level 5 Level 4 Level 3 Level 2 Level 1 Level 0 Level -1 Level -2 x3 x2 x3 x4 x5 x6 x2 x x2 x3 x4 x5 x P P 1 P Px x2 x3 x4

x2
x x2

x3 x2 x3 x4 x5 x6

x4 x3 x4 x5 x6 x7

x3
x4 x5

x7
x8 x9

x6
x7 x8

x5
x6 x7

x6
x7 x8

x7
x8 x9

x8
x9 x10

Value of Configuration
• Value of any configuration is taken to be the sum of values of the positions of all its pieces • A move can change the value of the config; so we x2 consider how a move changes the value • There are 3 types of...
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