When Florence and Willie finished playing darts, they proudly announced that their 3-digit scores added up to exactly 800 points. Furthermore, each of their scores shared the same 3 digits, without any repeating digits.
You could figure this out mathematically, but instead, I plugged in digits that added up to 10 for the first column, 9 for the second column (since you carry the 1) and 7 for the third column (again, because you’re carrying the 1).
A princess is as old as the prince will be when the princess is twice the age that the prince was when the princess’s age was half the sum of their present ages.
This one took a while to figure out and there are numerous valid ways of finding the answer.
Here is the solution I came up with
I created the following table from the riddle:
Current
Future
Past
Princess
x
2z
(x+y)/2
Prince
y
x
z
I then created three equations, since the difference in their age will always be the same. d = the difference in ages x – y = d 2z – x = d x/2 + y/2 – z = d
I then created a matrix and solved it using row reduction.
x
y
z
1
-1
0
d
-1
0
2
d
.5
.5
-1
d
It reduced to:
x
y
z
1
0
0
4d
0
1
0
3d
0
0
1
5d/2
This means that you can pick any difference you want (an even one presumably because you want integer ages). Princess age: 4d Prince age: 3d
Ages that work
Princess
Prince
4
3
8
6
16
12
24
18
32
24
40
30
48
36
56
42
64
48
72
54
80
60
To see other solutions check out the comments from when I posted this on my blog.
The name ABRAHAM can be changed into a word for a primitive musical instrument by replacing each letter with a different letter. The repeated letters (A is this case) must be replaced with the same replacement letter in the new word.
1. Each letter represents a different digit from 1 to 9 2. The total of each row is 17. 3. (B × B) + B + F = A 4. C × F = EF (a 2-digit number, not their product)
A + B + C = 17 D + E + F = 17 B2 + B + F = A C × F = EF
To begin with, B has to be a 1 or 2 or else A wouldn’t be a single digit. Plug in B = 2, gives you 6 + F = A, meaning F and A can only be (1,7) or (3,9). To get 17, C would have to be 8 or 6, but those values don’t work for C × F = EF. So B must be 1.
2 + F = A means F and A can be (2,4), (3,5), (4,6), (5,7), (6,8) or (7,9). To get 17 on the top row, the only option that leaves C as a single digit is F = 5 and A = 7.
C × F = EF 9 × 5 = 45, so E = 4 and D = 8 to make the second row equal to 17.
