Target Number and Properties of Zero, One and Exponents

My favorite math game by far is Target Number. I have modified the rules to make it more interesting and challenging. 

Preparation:  Take a deck of cards and separate out all of the face cards (eliminate Jokers, Jacks, Queens, and Kings). You should have 40 remaining cards from Ace to 10 in four suits. Ace counts as one. On page 5 of the pdf, I gave you a full deck of these 40 cards you can cut out (yes, they are very small).

1. Draw 5 cards face up.

2. Draw a 6th card face up

and place separately from the first 5; this is your target number.

3. Combine ALL 5 cards once using addition, subtraction, multiplication and/or division to result in the 6th card. You cannot use the 6th card in your calculations. There will be multiple methods to reach your solution. If your five cards are A, 4, 10, 8, and 7 and your target number was 8, one solution is: 

• 10 - 7 = 3 (now you have used the 10 and the 7 and have generated a 3)
• 4 - 3 = 1 (take the 4 card and subtract the number you generated above to get 1)
• 1 x 1 = 1 (take the Ace=1 card and multiply or divide it by the 1 you generated above)
• 1 x 8 = 8 (finally, take the last of the 5 cards, 8 and multiply by the 1 you generated above to generate your target number 8)

4. If you are the first one to find the solution, you keep the 6 cards.

5. The player with the most cards wins (since there are 40 non-face cards in a deck, there are 6 rounds in a single deck). Once the deck is finished, you can reshuffle and start again or keep a running total. You can play with more than two players or even play just by yourself. The biggest challenge is to find multiple solutions to each group of 6 cards. 

NOTE: Sometimes you will see the solution instantaneously, and sometimes, it may take you 15 minutes to find a solution (HINT: go into negative numbers).

Special hints: 

• First think of all combinations of numbers to get to your target number using + — x / so you can strategically think of ways to get to these numbers with the 5 numbers you have
• stay away from high numbers
• try to find the lowest numbers that will add, subtract or multiply to get your number
• try to get zero by subtracting any number from itself (a -- a = 0)
• try to get a one by dividing any number by itself (a / a = 1)
• add zero to your target number get the number added to; multiply by one to get the number multiplied by.

Random Card Generator: I set up a random card generator that would only choose 6 cards from 40 cards with J, Q, and K removed; let the last card be the Target Number:

https://www.random.org/playing-cards/?cards=6&decks=1&spades=on&hearts=on&diamonds=on&clubs=on&aces=on&twos=on&threes=on&fours=on&fives=on&sixes=on&sevens=on&eights=on&nines=on&tens=on&remaining=on

It is critical to know the properties of zero and one for any real number "a":

a + 0 = a a x 1 = a

a x 0 = 0 a  / 1 = a

0  / a = 0 (a ≠0) a  / a = 1 (a ≠0)

a  / 0 = undefined 1 x 1 = 1

Commutative Property:

a + b = b + a a x b = b x a

a - b NOT = b - a a / b NOT = b / a

With the 5-6th graders we looked at properties of exponents to use with Target Number. These included

a^0=1 (a≠0)

a^1=1

a^2=aa

a^3=aaa

a^(1/2)=the square root of a

a^(m/n)=the nth root of a^m

These wonderful properties take Target Number to the next level.

Please send me solutions to the many combinations of cards you attempt. Remember there are 810,000 combinations of cards you can draw. Are there any combinations without any solutions? One student suggested that the combination A A A A 2 with a target number of 10 had no solutions. The Mathletes and I only found one solution: 3^2 +1. Another student suggested that A A A A 2 with a target number of 9 had no solutions. We found about 5 solutions. This was such a fun challenge.