### Easy or Difficult?

by Albert Frank

Recently someone gave me the following problem:

On the Xth day of the Yth month of the year 1900+Z in the 20th century, a ship is near New York. The ship has T crew members, U propellers and V chimneys. If we add the cube root of the age of the captain (who is a grandfather) to the product XYZTUV, the result is 698823. What are the values of X, Y, Z, T, U, V, and what is the age of the captain? We also know that only one solution is realistic.

Is this problem difficult or easy? Let’s have a look at it (the solution):

The age of the captain (who is a grandfather) is a perfect cube: It can only be 64 years old.

698823 – 4 = 698819.

Let’s make a decomposition of 698819 into prime factors: 698819 = 11 x 17 x 37 x 101. We have four factors. Six are needed, so the two others are 1 and 1.

11 would be a too big a number for propellers or chimneys, so the ship has 1 propeller and 1 chimney.

The month has to be <13, so it can only be 11.

The day has to be < 32, so can only be 17.

The year (Z) has to be < 100, so can only be 37.

The remaining number 101 is the number of crew members.

We have it: 17th November 1937, 1 propeller, 1 chimney, 101 crew members, and the captain is 64 years old.

Some will find this problem very easy, others will find it very difficult.

## 2 comments:

Interesting problem. I worked out the age of the captain easily enough, but got stuck because I didn't know how to "decompose" a large number into prime factors.

I could see that the final '9' of 698819 could not be achieved by multiplying any even number, nor any number ending in 5. Of the others, *3x*3, eg. 23x13 might be possible; or *9, eg. 19; or *7x*7, eg. 17x37. A *3 can be composed of a *9x*7, and a *7 can be made from *3x*9. So various combinations of 3,7 and 9 seemed possible, with 1's to make up the remainder.

But beyond that I was done for, short of methodically trying out each prime number in sequence.

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