The mysterious alien box
An alien gave some scientists a mysterious box that has the number "401" written on its front. They managed to carefully open it and found exactly 85 objects. Considering that, they correctly concluded that the aliens use the number "401" to represent 85 units. The scientists used this information to discover the number of fingers those aliens have in both hands together. How many fingers they have?
Before opening the box, the scientists already knew that:
- The aliens use a number system very similar to the one used by humans.
- The aliens have two hands.
Tip: it is an integer value and can be odd.
The answer is 9.
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1 solution
Humans use the number of fingers they have in both hands (10) as the basis of their numeral system. For instance, let's analyze number 401 in the human system. The "4" means 4 hundred (4x100), 0 means 0 ten (0x10) and 1 means 1 unit (1x1). Each position a numeral holds in a given number has a different "weight" and each position has a weight of 10 times the previous position. It happens because humans decided to use this decimal system because it is much easier to use as it is based on the number of fingers humans hold in their both hands together (10 fingers).
The problem tells us that aliens also use a numeral system based on the number of fingers they hold in both hands. As we know they do not represent "85" as "85" we can conclude that their system is not based on ten, so they do not have 10 fingers.
We can write an equation to solve this saying that x is the number of fingers they have. If the number they use to represent 85 units is 401, let's try to consider that "1" holds the position of the unit, that "0" holds the position of the "ten" and that "4" holds the position of the "hundred". However, we already know that their "ten" does not represent 10 units because they do not have 10 fingers. Actually, it may represent the number of fingers they have in both hands. So, their "ten" values x. We also know that their "hundred" does not represent 100 units, but actually x times x, because the "hundred" position is defined by how many "tens" can be formed by groups of "tens". For instance, if the alien has 4 fingers and writes the number 100, they probably mean 16 units, because 16 is the same as 4x4: one group of "tens" (which means 4 in this case) of "tens", a group of 4 groups of 4 units. That is why the "hundred" position is defined by x^2.
Let's use this knowledge: in the number "401" probably the "4" means "4 hundred" (a "hundred" is x^2), the "0" means "0 ten" (a ten is x), and the 1 means "1 unit". Write the equation considering that this number represents 85 units:
85 = 4(x^2) + 0(x) + 1 84 = 4 (x^2) 21 = x^2
As you can see, the square root of 21 is NOT an integer number, but the problem says the answer is an integer number! So we may have made a mistake. This path will not lead us to the resolution.
Therefore, let's try to think in a different way! Be creative! We know that some cultures write from right to left, instead of the left to right as we do in English. Maybe it is the same with the aliens! The problem does not state how they write. Thinking this way, we can suppose that in the number "401" the "1" is holding the position of the "hundred", "0" is holding the position of the "ten" and "4" is holding the position of the unit. We already know that the "hundred" is x^2, the "ten" is x and that the unit is the same number that is stated. Using this reasoning we have:
85 = 1(x^2) + 0(x) + 4 81 = 1(x^2) + 0 81 = x^2 x = +9 OR x = -9
As we know that it is impossible to have a negative amount of fingers, we can ignore -9 and conclude that 9 is the correct answer. Therefore, the alien has exactly 9 fingers!
Now that you know the answer, challenge that friend of yours that always think that he/she is smarter than you!