two gems

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Two Mathematical Gems:

Two Mathematical Gems Daniel Ullman

Peg Solitaire

Peg Solitaire

Rules:

Rules One peg left in center: IQ=400, super genius! One peg left not in center: IQ=300, near genius Two pegs left: IQ=200, very smart Three pegs left: IQ=150, pretty good Four pegs left: IQ=100, not bad Five pegs left: IQ=75, try again Six pegs left: IQ=50, dumb as an eggplant Seven pegs left: IQ=25, dumb as a doornail Eight pegs left: IQ=3, dumb as a stick

Slide5:

Yellow: 11 Pink: 11 Green: 11

The key observation:

The key observation In the opening position, pegs fill 11 yellow holes, 10 pink holes, and 11 green holes. Every move changes the parity of the number of yellow holes filled, pink holes filled, and green holes filled. If only one peg is left at the end, it must be in a … pink hole!

Near genius?

The lion and the man:

The lion and the man A lion and a man are in a large cage. The maximum speed of the man is exactly equal to the maximum speed of the lion. Can the lion catch the man?

Huh?:

Huh? How big is the lion? How long can the man run? What shape is the cage? Who moves first? How frequently are decisions made? What is speed?

Man Lion

Man Lion

Lion Man

Slide15:

Lion Man 1 1/2 1/3 1/4 1+1/4+1/9

Slide16:

After n steps, the man has survived for 1 + 1/2 + 1/3 + 1/4 + . . . + 1/n minutes. The harmonic series diverges, so the man survives arbitrarily long. After n steps, the distance from the center of the circle to the man is the square root of 1 + 1/4 + 1/9 + 1/16 + . . . + 1/n2 furlongs. This series converges, so the man stays away from the edge of the cage. The man survives!

Good luck with this afternoon’s team competition!:

Good luck with this afternoon’s team competition!