Japanese Teahouse Plans

[The Problem of the week] A broken calculator

There goes the new problem

Imagine that your calculator has broken the key of "zero." The game is to appear on the screen these numbers: 250, 205, 2050, 0.025.

And what could make the following calculations? 0.025 · 205, 2050: 250.

Remember that the key of "zero" is broken.

gosh! Where is the solution? Do not panic ... Is lower after the illustrative image.

[Speaking of calculators, this strange thing with the name of Curt is in fact a mechanical calculator invented during World War II prisoner of a Nazi concentration camp, Curt Herzstark. Clearly, the Curta name comes from the name its inventor, who survived the concentration camp and perfected the design of the calculator, by pulling the market in 1948. Were considered the best hand calculators until 1970, when it began to be replaced by electronic calculators. The Curtas can add, subtract, multiply and divide, and by design were called pepper mills. You can see wikipedia page for more details and also simulator how ]

Solution:

is evident that this problem admits multiple solutions. For example, if you want to appear on the screen 250 without pressing the "zero" we can do any calculation that gives us 250 as a solution, and there are plenty of calculations in which we need not press the "zero": 249 + 1 251 to 1, 125 2, etc ...

Then I will write one of these infinite possibilities, looking for a few operations and a certain "elegance" in the choice of numbers, using mainly products and division instead of addition and subtraction:

250 = 125 • 2

205 = 41 • 5

2050 = 82 • 25

0.025 = 1: (8 ° 5)

Once we have these opportunities, just combine the two operations for asking us the problem:

0.025 · 205 = [1: (8 ° 5)] · 41 • 5

2050: 250 = (82 ° 25): (125 ° 2)

Enlargement:

Taking advantage of our solution we have used the products and quotients, the latter two operations can be simplified and get the same result with fewer steps.

0.025 · 205 = [1: (8 ° 5)] · 41 • 5, here we simplify a 5, which multiplied with another split, obtaining the same result as follows: 0.025 · 205 = 41 : 8

2050: 250 = (82 ° 25): (125 ° 2), we can simplify the 25 to 125 and 82 to 2, and the same result as follows: 2050: 250 = 41: 5

matenavegantes

When we find a problem that has many ways of being resolved, we are not satisfied with having found those solutions, but we wonder what solutions will be the shortest, most efficient, etc. Yes, we like to matenavegantes life difficult. In our case might be interesting to find out, for each of the operations, which is the minimum number of keys you have to click on our calculator to get the result.

Thus, for 250, it seems that we need at least five buttons plus clicking the "same", although that depends on the model of calculator you have.

If we make 250 = 125 ° 2 are five keys (more of the same, but from now on we will not tell), but if we do 250 = 5 ³ 2, and our calculator has a button that lifts the bucket directly, then just have to press four keys.

Similarly, 205 = 199 + 6, here are five keys to press, but the solution given above, 205 = 41 • 5, just press four.

I invite all readers to experiment with their own calculators and try to find those keys minimum numbers needed for each calculation.

Notes: This problem has been taken from the textbook publisher SM.