Using any or all of the operations of addition, subtraction, multiplication, division and exponentiation (and brackets), and as many of the number 1 as you need, produce each of the numbers from 2 to 20. For example, here is one way to make the number 17: $latex (1+1+1)^{(1+1)} \times (1+1) -1=17$ What is the smallest … Continue reading Only Ones

# 100!

# One Spot Dice

You have two blank 6-sided dice. You are allowed to choose any number of the twelve faces and draw exactly one dot on each of those faces. Now roll both dice. Can you draw the dots so that the possible totals are equally likely? Extension Put dots on several faces of two 6-sided blank dice. … Continue reading One Spot Dice

# Number of Letters

The numbers seven, eleven, fifteen, nineteen make a sequence where the numbers go up by 4 each time. But if you count the number of letters in each word, you get a sequence of numbers that goes up by 1 each time – 5, 6, 7, 8. The list above has 4 numbers in it. … Continue reading Number of Letters

# Lousy Labelling

Three boxes filled with lots of balls are on the table. One box is full of red balls, one is full of blue balls, and one is filled with both red and blue. Three labels are made for the boxes, but they are stuck to the wrong ones so that no box ends up with the right label. You can’t see what colour the balls in each box are unless you pull some balls out to look. How many balls do you need to pull out of the boxes to know which box is which?

# Combo Cube

Eight cubes are marked with one dot on two opposite faces, two dots on two opposite faces and three dots on two opposite faces. The eight cubes are glued together to form a bigger cube. The dots on each face of the large cube are counted, to get six totals. Can the cubes be arranged … Continue reading Combo Cube

# Convexity of Deltahedra

Claim: There is a convex deltahedron with every even number of faces from 4 to 20, except one. https://twitter.com/DavidKButlerUoA/status/1042234752453898240 What is the most the least convex deltahedron you can find? What is the most convex non-convex deltahedron you can find?

# Dividing Dice

A standard six-sided die spontaneously starts to divide like a living cell. The spots on the die move during the process and spread across the faces of the resulting pair of dice. That is, the two daughter dice share between them the spots from the original parent die, but possibly in new locations. After a … Continue reading Dividing Dice

# Broken Calculator

Imagine your calculator is broken. Although it will still display numbers, only the 4 key, + key, = key, and clear key work. Starting at 0, can you get the calculator to display 1000? What is the fastest way, with the fewest button presses?

# Four Dots, Two Distances

Find all configurations of four (distinct) points in the place that will determine exactly two distinct (non-zero) distances.

# Jiggly Numbers

A positive integer is said to be jiggly if it has four digits, all non-zero, and no matter how you arrange those digits you always obtain a multiple of 12. How many jiggly positive integers are there? Extension What about varying number of digits?