David Weber, Talei Kunkel, Alexander Martinez and, Rebecca Shultis Solutions for Chapter: Algebraic Expressions and Equations Puzzles and Tricks, Exercise 30: Unit review

Use algebra to justify why the following number trick always works.

$\bullet$Take an integer and add to it the next consecutive integer.

$\bullet$ Add $7$ to this result.

$\bullet$Divide your new number by $2$.

$\bullet$Subtract the original number.

$\bullet$The result is $4$.

Justify the number trick using algebra.

Take any three consecutive integers and add the highest and lowest number. Show that the sum is always double the middle number.

Justify the number trick using algebra.

Take any four consecutive integers and add the highest and lowest number. Show that the sum is always equal to the sum of the two middle numbers.

Justify the number trick using algebra. Take any five consecutive integers and add the highest and lowest number. Show that the sum is always double the middle numbers and the same as the sum of the other two middle values.

Find two numbers that differ by $5$, where the sum of the smaller one and four the larger one is $100$.

Find the three consecutive numbers such that three times the lowest one plus five times the middle one is equal to $39$ less than ten time the largest one.

You are creating a number trick of your own. You will ask a volunteer to select a number and then you will:

• ask them to multiply the number by two.

• choose an even number and ask your volunteer to add this number to the one in your head.

• say divide the result by 2 and subtract your original number.

What is the result? Justify your answer using algebra.

In pairs, try the following card trick to see if you and your partner can figure out how it works using algebra. Using a standard deck of $52$ playing cards, turn the top card of the deck face up. Begin counting from the face value of this card, turning card up from the deck until you have counted to $14$.