問72
頻出度★★☆
10進数の2，5，10，21を，五つの升目の白黒で次のように表す。
| 10進数 | 升目 |
|---|---|
| 2 | □□□■□ |
| 5 | □□■□■ |
| 10 | □■□■□ |
| 21 | ■□■□■ |
それぞれの升目が白のときは0，黒のときは升目の位置によってある決まった異なる正の値を意味する。この五つの升目の値を合計して10進数を表すものとすると，■■□□□が表す数値はどれか。
平成23年度 秋期 ITパスポート試験 問72
テクノロジ系基礎理論 › 離散数学
This question's explanation and your answers so far.
解説
問 72
Incorrectア
12 is 8 + 4, that is, the value for □■■□□ (the 2nd and 3rd cells black). It is the number you get if you misread the black cells as being shifted one position to the right, and it does not match the total for ■■□□□ (the 1st and 2nd cells black), which is 16 + 8 = 24.
Incorrectイ
20 is 16 + 4, that is, the value for ■□■□□ (the 1st and 3rd cells black). It amounts to mistaking the 2nd cell's value for 4, but the difference between 10 = □■□■□ and 2 = □□□■□ fixes the 2nd cell at 8, so this is wrong.
- ウ
正解 Work out the value of each cell position one at a time from the four examples given. Let v1 to v5 be the values of the 1st to 5th cells from the left when they are black. ・2 = □□□■□ ... only the 4th cell is black, so v4 = 2 ・10 = □■□■□ ... the 2nd and 4th cells are black, so v2 + v4 = 10 → v2 = 10 − 2 = 8 ・5 = □□■□■ ... the 3rd and 5th cells are black, so v3 + v5 = 5 ・21 = ■□■□■ ... the 1st, 3rd, and 5th cells are black, so v1 + (v3 + v5) = 21 → v1 = 21 − 5 = 16 In the pattern ■■□□□ that is asked about, the 1st and 2nd cells are black and the 3rd to 5th cells are white (= 0), so the total is v1 + v2 = 16 + 8 = 24, and the correct answer is “ウ 24”. Note that 5 = v3 + v5 can be split into 4 + 1, so the values of the five cells are 1, 2, 4, 8, 16 from the right — exactly the place weights of a binary number. In other words, this notation is itself a 5-digit binary number with black = 1 and white = 0, so reading ■■□□□ as 11000 (binary) = 16 + 8 = 24 gives the same answer.
Incorrectエ
30 is 16 + 8 + 4 + 2, that is, the value for ■■■■□ (the leftmost four cells black). In ■■□□□ the 3rd to 5th cells are white, so their values count as 0 and are not added; the 4 of the 3rd cell and the 2 of the 4th cell must not be included in the total.
ここがポイント
- In a notation where each position is assigned a fixed value, take differences between the known examples to pin down the value of each position one by one. Here 2 gives v4 = 2, 10 − 2 gives v2 = 8, and 21 − 5 gives v1 = 16.
- The values obtained are 1, 2, 4, 8, 16 from the right, which are the place weights of a binary number. Treating black as 1 and white as 0, the problem reduces to a radix conversion from a 5-digit binary number to decimal (■■□□□ = 11000 → 24).
用語