Earlier today I set these three puzzles, loosely based around 1729, the “taxicab number”. To read about the link between London cabs and 1729 please read the original post.
1. Square pair
What is the smallest number that can be expressed as the sum of a pair of squares in two different ways? Hint: it is less than 100
Solution
12 + 72 = 52 + 52 = 50
2. Strip tease
I have five strips of wood with lengths 1, 2, 7, 17 and 29 centimetres. It is impossible to arrange any three of these strips into a triangle. I would like to add another strip of wood, of maximum length 29cm, so that I still cannot take three strips and make a triangle.
How many different lengths are possible for the seventh strip, and what are they? Lengths must be a whole number of centimetres. As a bonus, what shape can I make using only these hypothetical extra strips?
CORRECTION: Previously this question had the length 9. That was a mistake. Also, no two lengths can be the same.
Solution 3, 4 and 5
In a triangle, the sum of the two smaller sides need to be larger the longest side. You can make a right-angled triangle with sides of length 3, 4 and 5.
3. Sick sixth
I have four numbers, a, b, c, and d. They may be whole numbers, or fractions.
There are six ways to multiply two of these numbers together: ab, cd, ac, bd, ad, bc.
The values of five of these products, but not necessarily in this order, are 2, 3, 4, 5 and 6.
What is the value of the sixth product?
Solution 2.4
If we multiply ab x cd or ac x bd or ad x bc we get abcd.
Thus there must be two pairs from 2, 3, 4, 5 and 6 whose product is the same.
The only possibility is 2 x 6 = 3 x 4 = 12. The sixth product x 5 = 12.
Thanks to George Watkins for today’s puzzles.
I’ve been setting a puzzle here on alternate Mondays since 2015. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.
