Square Dance

Geometry Level 1

A square dance has the following commands:

"Rotate Right" means everyone moves one pace around the square, counterclockwise.
"Rotate Left" means everyone moves one pace around the square, clockwise.
"Diagonal Swap" means the person in the lower-left position and the person in the upper-right position switch places, while the other two people stay still.

The Butterfly Basket is a dance that consists of many moves that individual do dancing in place, plus 21 repetitions of the sequence:

{Diagonal Swap -- Rotate Left -- Rotate Left -- Diagonal Swap -- Rotate Right -- }

If Amy starts the dance in the upper left corner of her square, where does she end up at the end of the Butterfly Basket?

The lower left corner The lower right corner The upper left corner The upper right corner

2 solutions

Jean David
Jul 17, 2016

diagonal swap = +180° 2 diagonal swaps= 0° 1 rotate left cancels 1 rotate right remains 1 rotate left !

Zandra Vinegar Staff
Oct 16, 2015

The sequence of 5 moves {DS -- RL -- RL -- DS -- RR} effectively amounts to a single Rotate Left. 4 Rotate Left commands amount to no net motion. Therefor 21 rotate left commands are the same as one. Therefore Amy winds up in the Lower Left Position in the square, one place clockwise of where she starts.

I agree with the first solution reasing, so the final result is a RL. But this means the final position is LOWER LEFT, and not as indicated

Francesc Reverter Sabaté - 4 years, 11 months ago

(RL is one place anti clockwise)

Francesc Reverter Sabaté - 4 years, 11 months ago

It is interesting to observe that if you assign the numbers 0..3 to the positions A..D, the operations can be expressed in arithmetic mod 4; the rotates are increment/decrement and the swap is multiplication by -1. Then the sequence of moves from position x leads to (-1 * x + 1 + 1) * (-1) - 1 = x - 2 - 1 = x + 1, which is a single Rotate Left.

Yury Kartynnik - 1 year, 9 months ago

In the same way, 21 of these gives 21 * 1 = 1 (mod 4).

Yury Kartynnik - 1 year, 9 months ago

But the single Rotate Left (or one place clockwise as you mention) from the upper left corner leads to the upper right corner, not the lower left marked as correct.

Yury Kartynnik - 1 year, 9 months ago

I agree...it leads to the upper right corner

Corazon Bautista - 1 year, 3 months ago

I agree I got that oo

Ann G - 3 months, 4 weeks ago

Yes one place clockwise will bring her to upper right, and not lower left

Atharva Vankundre - 5 months, 4 weeks ago

But the question states that the dance consists of the moves PLUS 21 repetitions... Which would be 22 sequences.

Christina Becker - 5 years, 3 months ago

It says: "consists of many moves that individual do dancing in place, plus 21 repetitions of the sequence", this is, the 21 repetitions are in addition to other dance moves, and not un addition to another initial repetition

José Manuel Barranco Gutiérrez - 5 years, 3 months ago

Square Dance

2 solutions

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