Teaser 2510

[Cactus]

Sunday Times Teaser 2510 by Danny Roth
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I have a four digit number. Writing down the number itself, the
average of its digits, the sum of its digits and the cube root of the
difference between the number and its (smaller) reverse, uses no digit
more than once. What is the number?

Cactus

[Ben Alligin]

This one is more like the standard I expect from Danny Roth, i.e.
easy, unlike the mirrors teaser which was, we suspect, more
complicated than the author imagined.

Anyway, if the number is abcd, then the difference between it and its
reverse is 999(a-d)+90(b-c). This is obviously divisible by 9 and has
to be a cube, so the only possibilities are 729, 1728, 3375 and 5832.
Therefore, 111(a-d)+10(b-c) = 81, 192, 375 or 648. It's pretty easy
after this, so I won't give my answer.

John

[Teasemaster]

Cubes which divide by 9 up to (9870-0789 = 9081) are 27, 216, 729,
1728, 3375, 5832
These must be expressible as 999(a-d)+90(b-c) which gives:
cube = 729, (a-d) = 1, (b-c) =+/-3
or 1728, (a-d) = 2, (b-c) =+/-3
The number so generated must divide by 4 (to produce an integer
average)
This gives us (1,3 condition):
1520, 1740, 1960, 2631, 2851, 3742, 3962, 4853, 5304, 5964, 6415,
7306, 7526, 8417, 8637, 9308, 9528, 9748
The only one where where the sum and average do not repeat digits is:
(number) 8417, (sum) 20, (ave) 5, (cubert diff) 10.8
BUT if (1,-3)
(number) 8147, (sum) 20, (ave) 5, (cubert diff) 9

I didn't check the (2,+/-3) condition

TM

[beermagnet]

I came at it with a much more sledgehammery approach thinking about
the possible values for the Sum and Ave.
Sum must be a multiple of 4, and Min,Max of SUM is 10,30 (for abcd
4321,9876)
So candidates for
Sum, Ave
12, 3
16, 4
20, 5
24, 6
28, 7
I quickly rejected all but
20, 5
24, 6
because for the others abcd must include one of their digits

Then looked at possible abcd for these:

abcd digits without 0,2,5
9731
9641
8741
7643

abcd digits without 2,4,6
9870
9753

For each of those six there are 12 possible combinations for abcd
where a>d
Tinkering around with them I fell upon the answer as given by TM
I was looking for cases where the diff was small because I guessed it
was more likely that the Cube and CubeRoot were small (single digit
CubeRoot) but didn't use the elegant logic shown by others above.
I have since checked that there are no other solutions.

Cheers, BM

[Ben Alligin]

I mentioned above that I thought the Danny Roth hadn't realised how
complicated the mirrors teaser was. Looking back, I noticed that it
was Danny Roth who wrote the train teaser (2472) where a boy is trying
to escape from an oncoming train. Again, that one turned out to be
more complicated than the author intended. Looking further back, I
have noted problems with Danny Roth's teasers on a number of
occasions, although these might well have been due to the sub-editors
at the Sunday Times rather than the fault of the author.

John

[Teasemaster]

Here's one to keep us going until Sunday...

Two men start work together, each paid £12,000 pa.
A gets a payrise of £600 every six months. B gets an annual payrise,
the same amount every year.
How much is B's annual payrise, if he ends up with the same salary as
A after 11 years?

TM

[Garry]

Hello TM

Taking "salary" to mean total earnings over 11 years, the equation
132000 + 55x = 201300 gives equality for the pair, whence B's annual
rise is x = £1260.

Garry

[Teasemaster]

"Salary" = annual earnings in one year
TM

[Ben Alligin]

If you mean that the salary after 11 years is the amount earned in the
11th year, then I make the annual pay rise £1230.

John