Triple Nine

Flow

Secret number. The first player enters a three-digit number (a number from 100 to 999) and tells her opponent that she is ready.

Cathy punches in her secret number 297, presses the plus key and says, “Ready!”

Add a number. The second player calls out a number that has one, two, or three digits. The first player adds this number to her original number. She then informs her opponent

• how many nines there in the total
• any other digit, but not its position

John says, “Add 123.” Cathy does so and gets a total of 420. She says, “No nines and a four.”

Repeat. Continue the process of the second player calling out a number, the first player adding it to to the total, and giving information about nines and one other digit. Keep track of how many moves the second player takes.

John says, “Add 555,” hoping to change the four to a nine regardless of its position. Cathy adds the 555 and gets 975. She tells John, “One nine and a five.” This is the end of the second move.

Four digits. If the total goes over 999 any time during the play of the game, the first player returns to the previous total. She says, “Overflow!” and does not give any additional information. This is counted as a move.

John guesses that the nine is in the hundreds place, though he is not sure of the location of the five. To make the maximum use of the situation, he asks Cathy to add 44. When Cathy does this, she gets a total of 1019. So she presses the minus key, cancels the last addition, and says, “Overflow.” This is the end of the third move.

John is not upset because he has collected useful information. His hunch about the nine in the hundreds place is confirmed. He also figures out that the tens digit (the digit in the middle) is greater than five because only then could the total have gone over a thousand. Therefore, the five must be in the units place. He calls out, “Add four.” Cathy's total is now 979 and she responds with “Two nines and a seven.”

John has figured out the entire number now. To finish off the game, he say, “Add 20.” Cathy does so and announces “Three nines.” The first game ends in five moves.

Reverse roles. The game is played again with the roles reversed. The second player in the previous game now selects a secret number and the other player tries to tun it up to 999.

Here's the complete game when it is Cathy's turn to guess:

Move 1. Cathy begins by saying, “Add 123.” John does so and announces, “No nines and a one.”

Move 2. Cathy figures out that the one cannot be in the hundreds place because John began with a three-digit number and added 123. So the “1” has to be in the tens or units place. Cathy guesses the latter and says, “Add an eight.” John does the addition and says, “No nines and a two.”

Move 3. Cathy takes a moment to process this information. Since she did not get a nine, the units digit was not the one. It must have been in the tens place. Since the units digit wasn't a one, adding eight would have made it ten or more, so the tens place must have gone up to 2. To clinch this digit, Cathy says, “Add 70.” John reports “One nine and a three.”

Move 4. Where is this three? Since this is the fourth round, Cathy does not think it is in the hundreds place. So she says, “Add six.” Her guess was wrong. John says, “Still one nine and a three.”

Move 5. Apparenly the three was (and still is) in the hundreds place. Cathy says, “Add 600.” As she expected, she gets two nines and a six.

Move 6. Cathy has the entire number now. She says, “Add 3,” and gets her triple nines.

Match. Two games make a match. The player who gets the triple nine with the least number of moves wins the match.

Since Cathy needed six moves and John only five, John wins.