Obsolete Machine 2
A machine is showing the number 0. It has 4 buttons: , , , and . If you press a button, the number that the machine is showing will be changed by the operation on the button. At least how many times do you need to press the buttons in order to make 2019?
The answer is 16.
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1 solution
It's easier to write in basis 2, as 2019 = 11111100011 in basis 2. Note that each operation can add an extra 1 digit, add or reduce the amount of digit, or change 100...00 into 011...10. That basically means, we can save some turns if there are several 1s grouped together. But since 2019 is an odd number, and there is no operation to create odd number other than dividing by 2, we definitely need to go through 4038 = 111111000110 basis 2. From here, our optimal step should be: 0 10 100 1000 10000 100000 1000000 10000000 1111110 11111100 111111000 1111110000 1111110010 11111100100 111111001000 111111000110 11111100011