## Teaser 2885: Croquet mallet

**From The Sunday Times, 7th January 2018** [link]

My croquet mallet is less than a metre high and consists of a cylindrical shaft attached to a heavier cuboid head, both of uniform density. Knowing the total weight of the mallet, I wanted to work out the weight of the shaft. I found the point of balance along the shaft and measured the distances from there to each end of the shaft, the smaller of which was less than the height of the head. Each of these three distances was a whole prime number of cm, and taking the three distances together with the height of the mallet, no digit was repeated. I worked out that the weight of the head was a single digit multiple of the weight of the shaft.

What was the height of my mallet?

[teaser2885]

## Jim Randell 11:29 am

on12 August 2019 Permalink |When the mallet is balanced we have:

where

a, b, care all primes andb < c < a.The total length of the mallet

t = a + b + c < 100.And between them, the numbers

a, b, c, thave no repeated digits.The shaft has a weight

wacting at a distance of(a + b)/2 – b = (a – b)/2from the balance point.The head has a weight

kwacting at a distance ofb + c/2.So for these to balance we have:

where

kis a single digit integer.This Python program finds the solution in 78ms.

Run:[ @repl.it ]Solution:The mallet is 90cm high.The values are:

t=90, a=83 b=2 c=5, k=9.LikeLike