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The main thing is a thousand puzzles of the week

2021-12-05T16:52:41.173Z

The number 1000 can be represented as the sum of four even numbers, but also as the sum of four odd numbers. Which representation are there more variants?



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Add up four numbers - you can probably do that without a calculator.

In the following exercise, however, they not only have to add but also combine.

There are two equations on the board.

In both of them there is the number 1000 on the left and a sum of four numbers on the right.

In the equation above, the four terms are all even.

In the equation below, they are odd.

Now there are quite a few different ways to represent the number 1000 as the sum of four even or four odd numbers.

The question is: of which representation are there more different variants?

(Note: The order of the summands does not matter. Swapping two summands does not lead to a new variant that differs from the original variant.)

There are more representations with four odd summands than with four even-numbered ones.

We look at the set of all even-numbered solutions (a, b, c, d).

The following applies to all:

1000 = a + b + c + d

The four summands should be sorted in ascending order of size for each individual solution.

In every even-numbered solution, the following applies:

d ≥ c ≥ b ≥ a

Now we change the four summands of all solutions (a, b, c, d) in the same way.

We subtract 1 from each of the first three summands, and add 3 to the fourth summand d.

This does not change the sum, but it now consists of four odd-numbered summands:

1000 = (a-1) + (b-1) + (c-1) + (d + 3)

We have now converted all even-numbered solutions (a, b, c, d) into an identically large number of odd solutions (a-1, b-1, c-1, d + 3).

We know that there must be at least as many odd solutions.

But there are even more odd-numbered solutions, because (a-1, b-1, c-1, d + 3) does not contain all possible variants.

Because d ≥ c, in all solutions of the form (a-1, b-1, c-1, d + 3) the fourth number (d + 3) is larger than the other three numbers.

We sorted a, b, c, d beforehand in ascending order of size.

But odd solutions are also possible, in which the fourth number is not the largest number, but instead, for example, has two largest summands of the same size.

Such as (249, 249, 251, 251).

There are also three equally large largest numbers possible such as (1, 333, 333, 333).

We have shown that in addition to the variants derived from all even-numbered solutions (a-1, b-1, c-1, d + 3) there are other odd solutions that are not included.

I discovered this ingenious riddle in the book "The Garden of the Sphinx" by Pierre Berloquin.

If you missed a puzzle from the past few weeks, here are the most recent episodes:

  • Heads or tails

  • Crazy

  • How big is the yellow triangle?

  • Which stick is magnetic?

  • Who will get the last coin?

  • Difficult legacy

  • How big is the circular ring?

  • Buying a bicycle with counterfeit money

  • Six acute angles

  • How far is it to the neighboring village?

Source: spiegel

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