Tuesday puzzle

Re: Tuesday puzzle

Bill Earl

138 seconds.

Re: Tuesday puzzle

Alex Y

Post your timeline, and let's see if anyone can beat it.

BTW, I forgot to mention, the timeline does have to include time to get the final pieces out of the toaster.

Time line

Bill Earl

The column on the left is seconds.

The toasters are labled 'a' and 'b'.

The slices are labled 1,2,3 & 4.

The 'flip sides' are indicated as 1', 2', 3' & 4'.

0 - load a 1

3 - load b 2

30 - remove a 1

33 - load a 3

36 - remove b 2

39 - load b 4

42 - butter 1

54 - butter 2

63 - remove a 3

66 - load a 1'

69 - remove b 4

72 - load b 2'

75 - butter 3

87 - butter 4

96 - remove a 1'

99 - load a 3'

102 - remove b 2'

105 - load b 4'

129 - remove a 3'

135 - remove b 4'

138 - done

I think I see the problem here!

Alex Y

The slices are labled 1,2,3 & 4.

From the original problem:

Your task is to prepare three [emph added] pieces of buttered toast

This time with 3 slices:

Bill Earl

The column on the left is seconds.

The toasters are labled 'a' and 'b'.

The slices are labled 1,2,3

The 'flip sides' are indicated as 1', 2', 3'

0 - load a 1

3 - load b 2

30 - remove a 1

33 - load a 3

36 - remove b 2

39 - butter 1

51 - load b 1'

54 - butter 2

63 - remove a 3

66 - load a 2'

69 - butter 3

81 - remove b 1 (done)

84 - load 3'

96 - remove a 2 (done)

114 - remove b 3 (done)

117 - complete

This is harder than it looks!

Alex Y

Your solution posted below with problems noted, followed by a adjusted solution.

The column on the left is seconds.

The toasters are labled 'a' and 'b'.

The slices are labled 1,2,3

The 'flip sides' are indicated as 1', 2', 3'

0 - load a 1

3 - load b 2

30 - remove a 1 Load was completed at t=3, so at this point, it has only toasted for 27 seconds

33 - load a 3

36 - remove b 2

39 - butter 1

51 - load b 1'

54 - butter 2

63 - remove a 3 only toasted for 27 seconds, and still buttering 2 for another three seconds

66 - load a 2'

69 - butter 3

81 - remove b 1 (done) only toasted for 27 seconds

84 - load 3'

96 - remove a 2 (done) only toasted for 27 seconds

114 - remove b 3 (done) only toasted for 27 seconds

117 - complete

Making only the obvious corrections, I would adjust your solution to :

0 - load a 1

6 - load b 2

33 - remove a 1

36 - load a 3

39 - remove b 2

42 - butter 1

54 - load b 1'

57 - butter 2

69 - remove a 3

72 - load a 2'

75 - butter 3

87 - remove b 1' (done)

90 - load 3'

105 - remove a 2' (done)

123 - remove b 3' (done)

126 - complete

Martin Gardner claims a solution of 120 seconds, and has had readers offer a 114-second solution.

120?

Bill Earl

0-3 - load a 1

3-6 - load b 2

33-36 - flip a 1

36-39 - remove b 2 (toasted one side)

39-42 - load b 3'

42-54 - butter 2 (buttered & toasted 1 side)

66-69 - remove a 1 (toasted both sides)

69-72 - load a 2

72-75 - flip b 3 (toasted one side)

75-87 - butter 1

- 1 done! @ 87

102-105 - remove a 2

- 2 done! @ 105

105-108 - remove b 3 (toasted both sides)

108-120 - butter 3

- 3 done! @ 120

  but can anyone do better?

Alex Y

I haven't seen Gardner's solution, but I got the same one as you.

Five of his readers came up with a solution using 114 seconds. To do so required eliminating an assumption that was implicit in the 120-second solution.

I will leave that challenge out there for a while...

Re:   but can anyone do better?

Bill Earl

"The first four operations above can only be performed one at a time (i.e., you can't be removing a slice from one side while simultaneously putting a slice in the other side)."



Strictly speaking, as the problem was stated: that doesn't rule out buttering a slice while the second side is toasting.

Re:   but can anyone do better?

Alex Y

I have to hand it to you, that is true, and would allow a much shorter time. However, that is not the implicit assumption that was the basis of the 114 solution.

I'll post the implicit assumption that is violated as a separate hint. Read if you want.

Hint for 114-second solution

Alex Y

Fully toasting one side of a piece of bread requires 30 not-necessarily-contiguous seconds in the toaster oven.

Re: Hint for 114-second solution

Larry Barrett

I got the same 120 sec solution, and was going to ask if the rules precluded interrupting one step to perform a second, like interrupting the 12 sec butter to do a 3 sec insert or flip. I guess from your hint that that is possible.

Yes

108

Bill Earl

Since the rules do not prohibit concurrent buttering & toasting:

0-3 - load a 1

3-6 - load b 2

33-36 - flip a 1'

36-39 - remove b 2

39-42 - load b 3

42-54 - butter 1

54-66 - butter 2

66-69 - remove a 1 (DONE!)

69-72 - load a 2'

72-75 - flip b 3'

75-87 - butter 3

102- 105 - remove a 2 (DONE!)

105-108 - remove b 3 (DONE!)

Alex Y

Bill gets the prize for the shortest time, with his creative buttering technique, which would be allowed by my paraphrase of the original problem (which had referred to the toaster as "the old fashioned kind" ). The original probably would have precluded this solution, as would I had I specified that the doors needed to be closed to toast. But as it is, this is a definite winner!

114-second solution

Alex Y

Bill wins the prize for his creative buttering technique.

If buttering can only be done outside of the toaster, the best that can be done is 114 seconds:

0-3 Put slice 1 in oven A

3-6 Put slice 2 in oven B

18-21 Remove slice 1 from oven A (toasted for 15 seconds)

21-24 Put slice 3 in oven A

36-39 Remove slice 2 from oven B

39-42 Put slice 1' in oven B

42-54 Butter slice 2

54-57 Remove slice 3 from oven A

57-60 Put slice 2' in oven A

60-72 Butter slice 3

72-75 Remove slice 1' from oven B

75-78 Put slice 3' in oven B

78-90 Butter slice 1'

90-93 Remove slice 2 from oven A

Done with 2

93-96 Put slice 1 in oven A

108-111 Remove slice 3 from oven B

Done with 3

111-114 Remove slice 1 from oven A

Done

Credit

Alex Y

This puzzle comes from My Best Mathematical and Logic Puzzles by Martin Gardner, 1994, Dover Publications