Can you find a five-digit number that has no zeros nor ones in it and no digit is repeated, where:
The fourth digit is a quarter of the total of all of the digits.The second digit is twice the first digit.The third digit is the largest.The last digit is the sum of the first two digits.
Reasoning
We can start by labelling the digits as ABCDE.
We know that:
(i) B = 2 x A
and:
E = A + B
And using (i) we get:
E = A + (2 x A) (ii) E = 3 x A
If A = 1, this isn't allowed (as there are no 1's in the puzzle).
If A = 2, then B = 4, and E = 6.
If A = 3, then B = 6, and E = 9, but this isn't allowed (as C has to be the largest digit).
So, A = 2, B = 4, E = 6, and we now have to find C and D.
We also know that:
D = (A + B + C + D + E) ÷ 4
And using (i) and (ii) we get:
D = [A + (2 x A) + C + D + (3 x A)] ÷ 4
so:
3 x D = (6 x A) + C
so:
(iii) D = [(6 x A) + C] ÷ 3
C can only be 7, 8 or 9 (as it's the largest digit, and we've already found 6) and (iii) tells us that it must be a multiple of 3, which means that C = 9. Leaving D = 7.
So the final number is: 24976.
Double-Checking
The answer is 24976.
The fourth digit is a quarter of the total of all of the digits.
A + B + C + D + E = 2 + 4 + 9 + 7 + 6 = 28, and 28 ÷ 4 = 7.
The second digit is twice the first digit.
4 = 2 x 2.
The third digit is the largest.
9 is the largest digit.
The last digit is the sum of the first two digits.
6 = 2 + 4.
Hint
There are 3 P's, so there are 3 different ways to start the word.
Answer
960.
Reasoning
Given: P P P U U Z Z Z Z Z L L L L E S S.
The order of the Z's is important. For example, we could chose a red one and a green one, so the order they appear in the sign has to be taken into consideration.
There are:
3 different P's that could be chosen.
2 different U's that could be chosen.
5 different Z's that could be chosen.
4 different Z's that could be chosen, as one has already been chosen.
4 different L's that could be chosen.
1 E that could be chosen.
2 different S's that could be chosen.
So there are 3 x 2 x 5 x 4 x 4 x 1 x 2 = 960 different ways to make the sign.
